A new vertex coloring heuristic and corresponding chromatic number
AA new vertex coloring heuristic andcorresponding chromatic number
Manouchehr Zaker ∗ Department of Mathematics,Institute for Advanced Studies in Basic Sciences,Zanjan 45137-66731, Iran
Abstract
One method to obtain a proper vertex coloring of graphs using a reasonablenumber of colors is to start from any arbitrary proper coloring and then repeatsome local re-coloring techniques to reduce the number of color classes. TheGrundy (First-Fit) coloring and color-dominating colorings of graphs are twowell-known such techniques. The color-dominating colorings are also knownand commonly referred as b-colorings. But these two topics have been studiedseparately in graph theory. We introduce a new coloring procedure whichcombines the strategies of these two techniques and satisfies an additionalproperty. We first prove that the vertices of every graph G can be effectivelycolored using color classes say C , . . . , C k such that ( i ) for any two colors i and j with 1 ≤ i < j ≤ k , any vertex of color j is adjacent to a vertex ofcolor i , ( ii ) there exists a set { u , . . . , u k } of vertices of G such that u j ∈ C j for any j ∈ { , . . . , k } and u k is adjacent to u j for each 1 ≤ j ≤ k with j (cid:54) = k ,and ( iii ) for each i and j with i (cid:54) = j , the vertex u j has a neighbor in C i . Thisprovides a new vertex coloring heuristic which improves both Grundy andcolor-dominating colorings. Denote by z ( G ) the maximum number of colorsused in any proper vertex coloring satisfying the above properties. The z ( G )quantifies the worst-case behavior of the heuristic. We prove the existence of { G n } n ≥ such that min { Γ( G n ) , b ( G n ) } → ∞ but z ( G n ) ≤ n . Foreach positive integer t we construct a family of finitely many colored graphs D t satisfying the property that if z ( G ) ≥ t for a graph G then G contains anelement from D t as a colored subgraph. This provides an algorithmic methodfor proving numeric upper bounds for z ( G ). AMS Classification:
Keywords:
Graph coloring; Coloring algorithms; Greedy coloring; Grundy color-ing; Color-dominating coloring ∗ [email protected] a r X i v : . [ c s . D M ] N ov Introduction
All graphs in this paper are simple and undirected. Let G be a graph with the vertexset V ( G ). By a proper vertex coloring of G we mean any function C : V ( G ) → N such that for each adjacent vertices u and v , C ( u ) (cid:54) = C ( v ). In this paper by vertexcolorings, we always mean proper vertex colorings of graphs. Denote by χ ( G ) theminimum number of colors used in any proper vertex coloring of a graph G . A verystrong result proved by Zuckerman [18] asserts that for every arbitrary and fixed (cid:15) > A such that for every graph G on n vertices, A ( G ) /χ ( G ) ≤ n − (cid:15) then N P ⊆ P , where A ( G ) denotes the numberof colors used by the algorithm A to color the graph G . This in particular provesthat it is a hard task to approximate the chromatic number of graphs within anyconstant factor. But graph coloring has wide applications in practical areas andhence exploring efficient graph coloring procedures is a necessary and vital goal.Efficient coloring heuristics is one of the important areas in graph theory and itsalgorithmic aspects (see e.g. [6, 14]). The greedy coloring procedure is one of thesimplest heuristics which has been widely studied in the literature. The on-lineversion of greedy coloring scans the vertices according to an ordering of vertices (i.e.on-line presentation of the graph), assign the color 1 to the first vertex and at eachstep color the current vertex by the smallest admissible color. In fact in any on-linecoloring, the information of any input graph G is presented gradually in on-line formi.e. vertex to vertex. A proper vertex coloring C has Grundy property if for anytwo colors i and j with i < j , every vertex of color j has a neighbor of color i . Anysuch coloring is called Grundy coloring. The Grundy coloring can be considered asthe off-line version of the greedy coloring, in the sense that the entire information ofthe coloring is presented in its definition. Note that this off-line version could alsobe considered as a color reducing technique, as follows. Name:
Grundy-type color reduction technique
Input:
A vertex coloring C of G with color classes C , C , . . . , C t Output:
A vertex coloring satisfying the Grundy property using at most t colors1. For i = 2 up to t
2. Do for any vertex v ∈ C i
3. If there exists j < i such that v has not any neighbor in C j ,let j ( v ) be the smallest such j
4. Do C i ← C i \ { v } C j ( v ) ← C j ( v ) ∪ { v }
5. If C i = ∅
6. Do remove C i from the listsand for each k > i rename C k by C k −
7. Else i ← i + 18. Return the refined color classes C , C , . . .
2t is easily seen that the output of the above technique is a proper coloring of G satisfying the Grundy property. Note also that every greedy coloring procedureoutputs a coloring satisfying Grundy property. Many of the coloring heuristics arespecial cases of greedy colorings in which an additional intelligent rule has beenapplied for choosing appropriate ordering of vertices. The greedy coloring obtainedfrom the smallest-last-order, the so-called Max-Degree-Greedy and DSATUR [4] aresome examples of this sort. The other famous heuristic is termed Iterated Greedy inwhich the greedy coloring is applied repeatedly with different orderings of vertices.We refer the readers to [5] and [6] for details and experimental properties of IteratedGreedy. All of these heuristics share a common property. The output of theseheuristics are colorings satisfying the Grundy property. The other important pointis that when these heuristics return output colorings (satisfying Grundy property)then no extra attempt is performed by the heuristics to reduce the number of colorsin the output colorings. In this paper we introduce a new coloring method to reducethe color classes obtained from any greedy coloring of an input graph.By a First-Fit (or Grundy) coloring of a graph G we mean any proper vertex coloringof G consisting of color classes C , . . . , C t such that for each i < j any vertex in C j has a neighbor in C i . The First-Fit chromatic number (or Grundy number) denotedby χ F F ( G ) (also by Γ( G )) is the maximum number of colors used in any First-Fitcoloring of G . The First-Fit colorings have been widely studied in graph theory (seee.g. [9, 17]). To determine the Grundy number is N P -complete even for complementof bipartite graphs [16]. In some cases greedy coloring is an optimal coloring [1].The complexity aspects of Grundy number was studied in many papers. See e.g.[3, 8, 10, 17]. In this paper we also use the concept of color-dominating coloring . Thecolor-dominating colorings are also known and commonly referred as b-colorings. Let C be a proper vertex coloring of G consisting of the color classes C , . . . , C t . Let C i be any color class in C . A vertex v ∈ C i is said to be a color-dominating vertex if foreach j (cid:54) = i there exists a neighbor of v of color j in C . Extracted from the conceptof color-dominating coloring we introduce the following color-dominating techniqueto reduce the number of color classes. Let C be any proper vertex coloring of G consisting of the color classes C , . . . , C t . Then the color-dominating technique actson the color classes of C and applies the following method to reduce the numberof classes. Take any class say C j from C . If C j contains a color-dominating vertexthen we keep the class C j unchanged in the coloring. Otherwise, for any vertex v of C j there exists a class C i such that v has no neighbor in C i . Remove v from C j andtransfer it to C i . By this method the class C j becomes empty and is removed fromthe list of color classes. Repeat this technique for all other classes. We eventuallyobtain a coloring in which every color class contains a color-dominating vertex. Thecolor-domination type technique is expressed in the following. Name:
Color-domination type color reduction technique
Input:
A vertex coloring C of G with color classes C , C , . . . , C t Output:
A color-dominating coloring using p ≤ t colors3. For j = 1 up to t
2. Do if C j has not any color-dominating vertex then resolve C j intothe other classes.3. For (cid:96) = j + 1 up to t rename the color class C (cid:96) by C (cid:96) − .4. Else the class C j remains unchanged, j ← j + 1 and go to the step 2.5. Go to the step 1 with the refined classes C , C , . . .
6. Return C , C , . . . For convenient we call a color-dominating vertex (resp. color-dominating class) aCD vertex (resp. CD class). A coloring C of G is color-dominating (or b-coloring) ifeach color class of C is a color-dominating class. The idea of b-coloring arises fromthe algorithmic approach to vertex coloring graphs. Let G be a graph and C , . . . , C k be any vertex coloring of G . For each i , if C i is not a color-dominating class thenwe can transfer each vertex of C i to a suitable class in C , . . . , C i − , C i +1 , . . . , C k .Hence by this method we can reduce the number of colors and eventually obtaina coloring which satisfies the b-coloring property. The maximum number of colorsused in any b-coloring of G is denoted by b ( G ) (also by ϕ ( G )) and is called theb-chromatic number of G . The b-chromatic number of graphs was firstly introducedand studied in [12]. Similar to Grundy coloring and number, the b-coloring andb-chromatic number of graphs have been the research subject of many articles. Seethe survey paper [13].For each positive integer t , a class of graphs denoted by A t was constructed in [17]which satisfies the following property. The Grundy number of any graph G is atleast t if and only if G contains an induced subgraph isomorphic to an element of A t .Any element of A t is called t -atom. The complete graph on one vertex (resp. twovertices) are the only 1-atom (resp. 2-atom). The elements of A t +1 are constructedby the elements of A t . For each t there exists only one unique tree t -atom denotedby T t . We have | V ( T t ) | = 2 t − and that T t is the largest t -atom. It was proved in[17] that there exists a function f ( t ) such that for every graph G on n vertices todetermine weather Γ( G ) ≥ t can be solved in at most f ( t ) n t − time steps. In otherwords, to determine the Grundy number of graphs is a problem belonging to thecomplexity class XP . For definition of fixed-parameter related complexity classessuch as XP , we refer the readers to the book [7]. The t -atoms have been used andstudied in some papers, e.g. [3, 8]. It was proved in [11] that the Grundy numberof trees can be determined in linear time. Let T be any tree and t any integer. Inorder to decide whether Γ( T ) ≥ t we have to check if T t is subgraph of T or not.For this purpose we can use the following result of Varma and Reyner in [15]. Let T and T be two arbitrary trees. There exists a polynomial time algorithm whichdetermines whether T is isomorphic to some subgraph of T . In other words, theSubtree Isomorphism is a polynomial time problem. The aim and outline of the paper are as follow.
The aim of this paper is to4ntroduce a new coloring heuristic which improves the greedy and color-dominatingcoloring heuristics. Because these coloring methods are commonly used in manycolor reducing algorithms, then they can be replaced by the new heuristic in orderto achieve more optimal colorings. We show infinitely many graphs for which the newheuristic is intensively optimal than their greedy and color-dominating colorings. Inthe next section we prove that the vertices of any graph G can be effectively coloredsatisfying three properties. This results in a new coloring heuristics and a newchromatic parameter denoted by z ( G ). It follows that z ( G ) ≤ min { Γ( G ) , b ( G ) } .In Section 3, we construct a family of “colored graphs” which we call z -atoms andprove that if z ( H ) ≥ t for some graph H and integer t , then H contains a z -atom G as “colored subgraph” such that z ( G ) = t . It implies that if H does not containany such z -atom then z ( H ) < t . This provides a polynomial time algorithm forproving upper bounds like z ( H ) ≤ t , where t is considered as a constant integer.In Section 4, we show that to determine z ( T ) for trees T is a polynomial timeproblem. The smallest tree T with z ( T ) = k has ( k − k − + k + 2 vertices. Forinfinitely many trees the new heuristic behaves extremely better than the Grundyand color-dominating colorings. In this section we first show that every graph admits a proper coloring which hassimultaneously Grundy and color-dominating properties. In the previous section wepresented the Grundy-type color reducing technique with details. This algorithmreceives an arbitrary proper vertex coloring of G and outputs a coloring satisfyingthe Grundy property. We summarize the algorithm in the following fact. Proposition 1.
Let C be any proper vertex coloring of a graph G . Then we cantransform C into a coloring which has Grundy property. Moreover, the transforma-tion is done in at most | E ( G ) | steps. Proof.
Let C , . . . , C k be the color classes in C . The Grundy-type color reducingprocedure transforms C into the desired coloring of G . It’s enough to determinethe time complexity of the algorithm. Note that when a vertex v is scanned andtransmitted to a class say C j then it will remain in C j during the rest of the algo-rithm, because in this situation v has a neighbor having color i , for each 1 ≤ i < j .Also, we need at most d G ( v ) (the degree of v in G ) operations to specify the class C j corresponding to the vertex v . Hence, the whole process needs at most 2 | E ( G ) | times steps. (cid:3) The next proposition achieves the first goal of this section.5 roposition 2.
Let C be any Grundy coloring of a graph G . Then we can transform C into a coloring which is simultaneously Grundy and color-dominating coloring.Moreover, the transformation is done in at most | E ( G ) | steps. Proof.
Let C , . . . , C k be the color classes in C . Any vertex in C k is adjacent tosome vertex from any other class. Hence, C k contains color-dominating vertex (orvertices). Let u be any vertex of C k − which has a neighbor in C k . Hence, C k − contains color-dominating vertices too. We consider now C k − . If it contains a color-dominating vertex then we go to check the class C k − . Otherwise, corresponding toeach vertex v of C k − there exists a smallest index i ( v ) (in this case i ( v ) ∈ { k − , k } )such that v is not adjacent to any vertex in C i ( v ) . Now, we move any vertex v from C k − to the class C i ( v ) . Note that C i ( v ) is upper than C k − and when a vertex ismoved to an upper class then it will not be scanned again in the rest of procedure.After removing all vertices of C k − then it becomes empty. We decrease by one unitthe color of any vertex in C k − ∪ C k . Denote the resulting new coloring by C (cid:48) . Notethat C (cid:48) has Grundy property and uses k − C (cid:48) k − , C (cid:48) k − in C (cid:48) contain color-dominating vertices. By continuing this method, assume thatwe have a Grundy coloring of G consisting of the classes D , . . . , D t such that each D i with i ∈ { t, t − , . . . , j + 1 } contains at least one color-dominating vertex. Weexplain how to handle the class D j . If D j has a CD vertex then we go to check theclass D j − . Otherwise, for each vertex v of D j there exists a smallest index p ( v ) (inthis case p ( v ) ∈ { j + 1 , . . . , k } ) such that v is not adjacent to any vertex in D p ( v ) .Now, we move any vertex v from D j to the class D p ( v ) . Then the class D j becomesempty. We decrease by one unit the color of any vertex in D j +1 ∪ . . . ∪ D k and replacethe coloring D by the resulting new coloring D (cid:48) . Note that D (cid:48) has Grundy property.The above procedure either eliminates a color class from the underlying coloring orfinds a color-dominating vertex belonging to the same class. Since the elimination ofclasses can not occur more than | V ( G ) | times then we eventually obtain a Grundycoloring which is also a color-dominating coloring. Also each vertex is scanned atmost once in this procedure and for each vertex v the number of steps to checkthe vertex v is at most d G ( v ). Therefore, the whole procedure can be done in (cid:88) v ∈ G d G ( v ) = 2 | E ( G ) | time steps. (cid:3) We summarize the procedure explained in the proof of Proposition 2 as the followingheuristic. The proof shows that the heuristic returns a coloring which is Grundyand color-dominating.
Name:
Grundy Color-Dominating Heuristic
Input:
A graph G and an arbitrary Grundy coloring C of G with say k colors Output:
A proper coloring C (cid:48) of G which is simultaneously Grundy and color-dominating coloring, where at most k colors are used1. C , C , . . . are color classes in a coloring C satisfying Grundy property2. k := the number of classes in the coloring C
6. For j = k − C j has a CD vertex then j ← j −
15. Else for each v ∈ C j ,6. Do C i ( v ) ← C i ( v ) ∪ { v }
7. For each i > j , C i ← C i −
8. Refine the color classes C , C , . . . and go to step 19. Return the final refined classes C , C , . . . Proposition 2 shows that every graph admits a color-dominating coloring satisfyingGrundy property. Let G be a graph and C be a Grundy and CD (i.e. color-dominating) coloring for G using k colors. There exist at least k CD vertices withdifferent colors in C . The subgraph of G induced on these color-dominating verticesmight be non-connected in general. This causes some problems in algorithmic anal-ysis of such colorings, because in this case any k vertices are potentially CD verticesand we have to explore all (cid:18) | V ( G ) | k (cid:19) subsets to obtain k color-dominating vertices.In Theorem 1, we show that we can transform any Grundy and CD coloring C of G using t colors to another Grundy and CD coloring C (cid:48) using say k colors with k ≤ t and satisfying the additional property that C (cid:48) contains a set D of color-dominatingvertices with different colors such that G [ D ] is isomorphic to a star graph i.e. K ,k − . Definition 1.
By a z -coloring of a graph G we mean any proper vertex coloring C of G using say k colors which is simultaneously Grundy and color-dominatingcoloring and contains a set { u , . . . , u k } of color-dominating vertices such that foreach j , the color of u j is j and for each j (cid:54) = k , u k is adjacent to u j . Denote by z ( G ) ( z -number of G ) the maximum number of colors used in any z -coloring of G . In Theorem 1, we prove that every graph G admits a z -coloring. In order to provethe theorem we need the concept of nice vertex. Definition 2.
Let C be any proper coloring of G using t colors. A vertex v is calleda nice vertex for the coloring C if the color of v is t in C and v is adjacent to atleast t − color-dominating vertices with t − different colors in C . Theorem 1.
Let G be a graph on n vertices and m edges. Let C be any Grundyand color-dominating coloring for a graph G using t colors. There exists an O ( mn ) procedure which transforms C into a Grundy and CD coloring C (cid:48) using say k colorssuch that k ≤ t and C (cid:48) contains a set { u , . . . , u k } of color-dominating vertices suchthat for each j the color of u j is j (in C (cid:48) ) and u k is adjacent to u j for each j (cid:54) = k . Proof.
Let C t be the class of vertices with color t in C . Since C is Grundy then eachvertex in C t is a CD vertex. If the class C t contains a nice vertex then C satisfies7he conditions of proposition. Otherwise, corresponding to each vertex u ∈ C t thereexists a minimum color i ( u ) such that u is not adjacent to any CD vertex of color i ( u ). It follows that for each vertex u ∈ C t and for each vertex w ∈ N ( u ) of color i ( u ) in C , there exists a smallest color j ( w ) such that j ( w ) is not appeared in theneighborhood of w . We have i ( u ) < j ( w ) < t . Note that to check if u is nice vertexis done in O ( m ) time steps. During this check and in case that u is not nice vertexthe values i ( u ) and j ( w ) are also determined for each w ∈ N ( u ). Let now u be anarbitrary vertex of C t . We make a local recoloring as follows. Change the color of u from t to i ( u ) and assign the color j ( w ) to any neighbor w of u whose colorin C is i ( u ). Observe that after this recoloring the vertices u and its neighborsremain Grundy vertices. If necessary using the techniques of Propositions 1 and 2modify the color of some vertices in C and obtain a Grundy and CD coloring C (cid:48) .Note that either there exists no class of color t in C (cid:48) or the number of vertices ofcolor t in C (cid:48) is strictly less than | C t | . We repeat this technique to the other verticesof color t in C (cid:48) . Such possible vertices of color t in C (cid:48) are already of color t in thecoloring C . Either we obtain a nice vertex in the underlying coloring and thereforethe procedure is finished at this step or we obtain a Grundy and CD coloring C (cid:48)(cid:48) which uses less than t colors. Again, if C (cid:48)(cid:48) has a nice vertex then C (cid:48)(cid:48) satisfies theconditions of the proposition and proof completes. Otherwise, let C (cid:48)(cid:48) p be the classof colors with the maximum color in C (cid:48)(cid:48) . Instead of C and C t we repeat the sametechnique for C (cid:48)(cid:48) and C (cid:48)(cid:48) p . By continuing this method either we obtain a coloringwith a nice vertex or one color class is removed and we obtain a Grundy and CDcoloring with fewer colors. Obviously the color classes can not be removed morethan | V ( G ) | = n times. We conclude that eventually a coloring C (cid:48)(cid:48)(cid:48) satisfying thedesired conditions is achieved.Obviously, the number of total iterations is at most n . By Propositions 1 and 2,each iteration needs time complexity O ( m ). As explained before, to check if anindividual vertex is nice takes O ( m ) steps. Hence the entire process of exploringnice vertices is done in O ( mn ) time steps. Overall, the transformation of C into C (cid:48)(cid:48)(cid:48) is performed in O ( mn ) time steps. This completes the proof. (cid:3) In the light of Proposition 2 and Theorem 1 we introduce a new coloring heuristicand an associated chromatic parameter. By the z -coloring heuristic we mean thecoloring heuristic explained as follows. By the proof of Theorem 1, the followingheuristic outputs in O ( mn ) time steps a z -coloring on any graph with n vertices and m edges. Name: z -coloring heuristic Input:
A graph G Output: A z -coloring of G satisfying Definition 11. Start from any proper vertex coloring C of G
2. Apply the Grundy technique and Grundy Color-Dominating heuristic for C and obtain a Grundy and color-dominating coloring consisting of the color classes8 , C , . . . t := the number of colors in C
4. For each vertex u ∈ C t
5. Do if u is nice vertex then go to step 11 and return C
6. Else, determine a minimum color i ( u ) such that u is not adjacentto any CD vertex of color i ( u )7. For each vertex w ∈ N ( u ) of color i ( u ) in C
8. Do determine a smallest color j ( w ) such that j ( w ) is notappeared in the neighborhood of w
9. Do c ( u ) ← i ( u ) and c ( w ) ← j ( w )10. Refine the color classes and go to the step 211. Return the final classes C , C , . . . We have the following proposition.
Proposition 3.
For any graph G , z ( G ) ≤ min { Γ( G ) , b ( G ) } . Moreover, there existsan infinite sequence of graphs { G t } ∞ t =1 such that Γ( G t ) → ∞ and b ( G t ) → ∞ as t → ∞ but for each t , z ( G t ) = 3 . Proof.
The inequality clearly holds. For each positive integer t , define H t = K t,t \ ( t − K . It is easily seen that Γ( H t ) = t + 1. It is also observed that H t doesnot admit any b-coloring using more than 2 colors. Construct another sequence ofgraphs as follows. Let t be any arbitrary and fixed positive integer. First considera path on t vertices on the vertex set { v , . . . , v t } . Then attach t − v and additional t − v t . Also, for each i with 2 ≤ i ≤ t −
1, attach t − v i . All of these leaves are distinct. Denote the resulting graph by F t . Weclaim that b ( F t ) = t . For each i , assign color i to the vertex v i . Now, using the leafvertices we can extend this pre-coloring to a b-coloring of F t using t colors. Since∆( F t ) = t − b ( F t ) ≤ ∆( F t ) + 1 then b ( F t ) = t . Now, consider the graph H t in one side and the graph F t in other side. Let u be any vertex of degree t in H t .Put one edge between the vertex u from H t and the vertex v from F t and thensubdivide this edge by putting an additional vertex say w . Denote the resultinggraph by G t , where there exists a path of length two between u and v . The graph G is depicted in Figure 1. Obviously, Γ( G t ) → ∞ and b ( G t ) → ∞ as t → ∞ . Tocomplete the proof we argue that z ( G t ) ≤ t . Its proof is simple. Assumethat G t admits a z -coloring using four or more colors. Let v be a vertex of color4 in a z -coloring of G t . The vertex v needs at least three neighbors of degree atleast three. Hence v (cid:54)∈ V ( F t ). Also v can not be w because the degree of w in G t istwo. Therefore v should be a vertex in H t . As we said v needs three neighbors ofdegree at least three. None of such neighbors of v can be w because w has degreetwo. It follows that every z -coloring of G t using more than three colors reduces toa b-coloring with more than three colors in H t . But as explained before this is notpossible. (cid:3) G t in Proposition 3 for t = 4A graph parameter p is said to be monotone if for every graph G and any inducedsubgraph H of G , p ( H ) ≤ p ( G ). An interesting property of z -coloring is that it’snot monotone, i.e. if H is an induced subgraph in G then it does not necessarilyimply that z ( H ) ≤ z ( G ). For example let G = K t +1 ,t +1 \ tK and H = K t,t \ tK .Then H ⊆ G , z ( G ) = 2 but z ( H ) = t . This is interesting because suppose that agraph G has a large z ( G ) and let C be a z -coloring of G using z ( G ) colors. Thisdoes not provide a good upper bound for the ordinary chromatic number of G , butthere is still a chance that if you add one vertex u (and in some cases two adjacentvertices u, u (cid:48) ) to G and u (resp. u, u (cid:48) ) becomes a color-dominating vertex for C thenthe z -number reduces significantly and therefore the graph properly colored with ansmaller number of colors. We present some examples. Let P be the path on fivevertices v , . . . , v , where v , v are its leaves and v its central vertex. Assign color 3to v , color 2 to v , v , and assign color 1 to v , v . It follows that z ( P ) = 3. Add anew vertex u and put edges between u and v , v , v . Assign color 4 to u and obtain aGrundy coloring of the new graph using 4 colors. Apply the technique of Proposition2 to these data and obtain a 2-coloring for the whole graph and hence for the P .The second example is the cycle on 6 vertices C . Color its vertices consecutively3 , , , , , z -coloring using 3 colors. A main difference between thesetwo examples is that in the case of C all vertices are color-dominating. In thissituation we add two adjacent vertices u, u (cid:48) to C and put enough edges between u, u (cid:48) and C such that they become color-dominating and share no common neighbor.Assign colors 4, 5 to u, u (cid:48) , respectively. Denote the resulting graph by H and coloringby C . The coloring C is Grundy coloring using 5 colors. Apply the technique ofProposition 2 for ( H, C ). If H is bipartite then we obtain a proper 2-coloring for H and hence for C . Otherwise, we obtain a 3-coloring for C . Note that the graphs G = K t +1 ,t +1 \ tK and H = K t,t \ tK satisfying z ( G ) = 2 but z ( H ) = t (mentionedin the proof of Proposition 3 and at the beginning of this paragraph) have propertiessimilar to our second example. In fact the graph G is systematically obtained from H using this technique, i.e. two adjacent vertices are added to H and so on.Based on the preceding examples and comments we add an important extra step toour proposed heuristic to achieve a complementary from of the heuristic. The complementary form of the heuristic: z -coloring heuristic has colored a given graph G and output thecolor classes C , . . . , C t . Denote the Cartesian product of these color classes by C × C × . . . × C t . For any element ( v , v , . . . , v t ) ∈ C × C × . . . × C t , definea graph denoted by G ( v ,v ,...,v t ) as follows. Corresponding to ( v , v , . . . , v t ) add anew vertex to G and connect the new vertex to the vertex v i , for each i = 1 , . . . , t .For each element ( v , v , . . . , v t ), apply the z -coloring heuristic to G ( v ,v ,...,v t ) . Eachresulting coloring comprises a vertex coloring of G . Choose the one with minimumnumber of colors. z -atoms and their properties In order to explain what we accomplish in this section, we need to introduce aconcept.
Definition 3.
Let H and G be two graphs. Let C be a proper vertex coloring of H . For each vertex v of H , denote the color of v in H by C ( v ) . We say ( H, C ) isembedded in G if there exists an injective function f : V ( H ) → V ( G ) such that thefollowing two conditions hold:(i) for each two vertices u, v of H if uv ∈ E ( H ) then f ( u ) and f ( v ) are adjacent in E ( G ) (i.e. H is isomorphic to a subgraph of G )(ii) if C ( u ) = C ( v ) (i.e. u, v have identical colors in H ) then f ( u ) and f ( v ) are notadjacent in E ( G ) . For example, let H be the path on four vertices and C be a Grundy coloring of H using three colors. Let G be the cycle on four vertices. Then H is subgraph of G but ( H, C ) is not embedded in G .In the following for each positive integer t we construct a collection of graphs D t such that each member of D t is a colored graph such as ( G, C ), where C is a z -coloring of G using t colors. The coloring C is called the canonic z -coloring of G . We prove that D t has the following property. Let L be any graph such that z ( L ) ≥ t . Then there exists ( G, C ) in D t such that ( G, C ) is embedded in L . As weexplained in the paragraph before Section 3, we are lucky that the converse of thisresult does not necessarily hold. In the following we say a graph H is edge-minimalwith respect to z -number (or simply edge-minimal) if for any edge e of H we have z ( H \ e ) ≤ z ( H ) −
1. Let S ,t be the star graph consisting of a central vertex ofdegree t and t leaves u , u , . . . , u t adjacent to the central vertex. We assign the color t + 1 to the central vertex and the color i to the vertex u i , for each i ∈ { , , . . . , t } .Let t ≥ D t +1 during two distinctphases. Starting from the very star graph S ,t , in Phase I we generate all possible11raphs H together with a proper vertex coloring C for H satisfying the followingproperties • The graph H contains S ,t as subgraph and for any p ∈ { , , . . . , t } , the colorof u p in C is p . Moreover, for any p with 1 ≤ p ≤ t − (cid:96) with p + 1 ≤ (cid:96) ≤ t , the vertex u p has a neighbor of color (cid:96) in H . ♣ • For any edge e of H , H \ e does not satisfy ♣ .All colored graphs ( H, C ) produced in Phase I, enter Phase II. In Phase II a graph(
H, C ) is extended to a collection of colored graphs each of which is typically denotedby (
G, C (cid:48) ) such that (
H, C ) is embedded in G and the restriction of C (cid:48) to the verticesof H is identical to C . Moreover, C (cid:48) is a z -coloring of G using t + 1 colors and G is edge-minimal graph with respect to z -number, where the vertices of S ,t are thecolor-dominating vertices with colors t + 1 , t, . . . , ,
1. Finally, the collection D t +1 is defined as the collection of all such graphs ( G, C (cid:48) ) produced in Phase II. In fact,the Phase II produces all edge-minimal colored graphs with z -number t + 1 andcontaining S ,t . In terms of graph notations, the following sequence denotes theevolvement of our constructions. S ,t −→ ( H, C ) −→ ( G, C (cid:48) ) . In Theorem 2, we prove that for every graph L if z ( L ) ≥ t then there exists amember ( G, C (cid:48) ) from D t such that ( G, C (cid:48) ) is embedded in L . Construction Phase I:
We consider the star graph S ,t with leaf vertices u , u , . . . , u t and its coloringusing t + 1 colors and want to generate all possible edge-minimal graphs ( H, C )satisfying the above-mentioned conditions ♣ . In fact, for any p and (cid:96) ≥ p + 1,the vertices u , . . . , u p need neighbors (in H ) with color (cid:96) (in C ). These neighborsare not necessarily distinct. It follows that, for each j with 2 ≤ j ≤ t we shouldhave j − j in ( H, C ). Hence, we needpotentially say m j distinct vertices of color j in ( H, C ), where m j can be any valuewith 1 ≤ m j ≤ j −
1. Note that we have already a vertex of color j in the star graph,i.e. the vertex u j . This means that one possibility is that all vertices u , . . . , u j − are adjacent to u j in order to have a neighbor of color j . But this is only one ofthe possibilities. Another important point is that if we satisfy the above-mentionedconditions for the neighbors of u , u , . . . , u t then the task is finished since becauseof minimality we need no extra vertices or edges. Based on this background, weexplain Phase I gradually as follows:For any j , 2 ≤ j ≤ t , let m j be an arbitrary integer with 0 ≤ m j ≤ j −
1. If m j = 0then define M j, = { u j } . If m j ≥ M j,m j be a set consisting of u j and12 j additional vertices. We have | M j,m j | = m j + 1 and for any j, j (cid:48) with j (cid:54) = j (cid:48) , M j,m j ∩ M j (cid:48) ,m j (cid:48) = ∅ .Corresponding to any fixed selection of m , m , . . . , m t and any set of surjectivefunctions { f j,m j : j = 2 , , . . . , t } , where f j,m j : { , , . . . , j − } → M j,m j define agraph H as follows.Add the vertices of M ,m ∪ M ,m ∪ . . . ∪ M t,m t as all distinct and extra vertices to S ,t . Note that if m j = 0 for some j , then no vertex corresponding to j is added tothe graph S ,t because as specified before, we have M j, = { u j } and u j is alreadypresent in S ,t (and therefore in H ). Now, for each j ∈ { , . . . , t } and any value p ,1 ≤ p ≤ j −
1, put an edge between u p and f j,m j ( p ). Assign the color j to all verticesin M j,m j .Denote the resulting graph and vertex coloring by H and C , respectively. Observethat for any p and any (cid:96) ∈ { p + 1 , . . . , t } , the vertex u p has a neighbor of color (cid:96) in H . Because, u p is adjacent to f (cid:96),m (cid:96) ( p ) and the color of f (cid:96),m (cid:96) ( p ) is (cid:96) in the coloring C (in fact f (cid:96),m (cid:96) ( p ) ∈ M (cid:96),m (cid:96) ).Before we proceed toward the second phase of construction we prove a result con-cerning the collection of graphs generated in Phase I. Proposition 4.
Let G be any graph and C be a z -coloring of G using t + 1 colors.Then there exist t +1 vertices u , u , . . . , u t +1 of colors , , . . . , t +1 in C , respectivelyand a colored graph ( H, C (cid:48) ) satisfying the following properties.(i) For any i , u t +1 is adjacent to u i .(ii) For each i , ≤ i ≤ t + 1 , u i is a color-dominating vertex of color i .(iii) Started from the star graph consisting of the vertices u , u , . . . , u t +1 , the graph ( H, C (cid:48) ) is subgraph of G and one of the graphs generated in Phase I, where thecoloring C (cid:48) is the restriction of C to the vertices of H . Proof.
Since C is a z -coloring then by the definition there exist t + 1 color-dominating vertices say u , u , . . . , u t +1 in G which satisfy the conditions ( i ) and ( ii ).Since u , u , . . . , u t +1 are color-dominating vertices then corresponding to any value j with 2 ≤ j ≤ t there exists a set M j of vertices having color j in C such that forany p with 1 ≤ p ≤ j − u p is adjacent to some vertex in M j . The set M j can be easily chosen so that each vertex of which has a neighbor among u , . . . , u j − .Note that the vertex u j itself may or may not belong to M j . We consider the sub-graph H of G on the vertex set { u , u , . . . , u t +1 } ∪ M ∪ M ∪ · · · ∪ M t and consistingof the above-mentioned edges between { u , u , . . . , u t +1 } and M ∪ M ∪ · · · ∪ M t aswell as the edges among u , . . . , u t +1 . Let C (cid:48) be the coloring of H obtained from therestriction of C to the vertices of H . Considering the sets M j and functions whichmap any vertex from { u , . . . , u j − } to its neighbor in M j , we conclude that ( H, C (cid:48) )is one of the graphs produced in Phase I. This completes the proof. (cid:3) onstruction Phase II: We need first to define the concept of Grundy class. Let H be an arbitrary graphand C a proper vertex coloring of H consisting of the color classes C , . . . , C t . Acolor class say C j is said to be Grundy class if for any color i < j any vertex u of C j has a neighbor of color i in H . In the following we introduce the operation G k ,where k is an arbitrary natural number. The operation G k operates on arbitrarycolored graphs ( H, C ) and outputs a collection of colored supergraphs of H . Let thecolor classes in ( H, C ) be C , . . . , C k , . . . , C t . There are two possibilities concerning C k . If the class C k is Grundy then G k operates on ( H, C ) and outputs the samegraph, i.e. G k ( H, C ) = (
H, C ) in this case. If C k is not a Grundy class then theoutput of G k is a collection of colored graphs typically denoted by ( G, C (cid:48) ) such thatthe following hold. The graph H is an induced subgraph of G , the restriction of C (cid:48) to the vertices of H is the same as the coloring C , both colorings C and C (cid:48) have thesame number of colors and the k -th class in C (cid:48) is a Grundy class. We may interpretthat the operation G k “Grundify” the color class k in its input graphs ( H, C ). The operation G k : Let (
H, C ) be an arbitrary input graph for the operation G k , where C k is not aGrundy class in ( H, C ). Let i ∈ { , . . . , k − } be an arbitrary and fixed integerand let C ik be a subset of C k consisting of the vertices with no neighbor of color i in C . Recall that C i is the set of all vertices having color i in C . Denote anarbitrary subset of C ik by S i . Let S i be an arbitrary and fixed subset of C ik . Let f i be any arbitrary and fixed function from S i to C i . Let also m i be any arbitraryvalue with 1 ≤ m i ≤ | C k \ S i | . Consider some extra vertices w i , . . . , w im i and let g i be any arbitrary surjective (onto) function from C ik \ S i onto { w i , . . . , w im i } . Set S = ( S , . . . , S k − ), f = ( f , . . . , f k − ), m = ( m , . . . , m k − ) and g = ( g , . . . , g k − ).Now corresponding to ( S , f , m , g ) we construct a graph H S , f , m , g as follows.Recall that corresponding to each i , 1 ≤ i ≤ k − S i ⊆ C ik , afunction f i : S i → C i , an integer m i with 1 ≤ m i ≤ | C k \ S i | and finally a surjectivefunction g i : C ik \ S i → { w i , . . . , w im i } . Now, corresponding to each i we perform thefollowing operations. Add the m i extra vertices w i , . . . , w im i to the graph H and putan edge between any vertex u ∈ C ik \ S i and g i ( u ). Note that g i ( u ) ∈ { w , . . . , w m i } .Next, we put an edge between any vertex u (cid:48) ∈ S i and f i ( u (cid:48) ). Note that f i ( u (cid:48) ) ∈ C i .Denote the resulting graph by H S , f , m , g . We have V ( H S , f , m , g ) = V ( H ) ∪ ( k − (cid:91) i =1 { w i , . . . , w im i } ) . Define a proper vertex coloring C (cid:48) of H S , f , m , g as follows. For any vertex v ∈ H set C (cid:48) ( v ) = C ( v ) and for each j ∈ { , . . . , m i } , C (cid:48) ( w ij ) = i . Note that in H S , f , m , g theclass of vertices of color k is a Grundy class. Remark 1.
Let ( H, C ) be any typical graph constructed in Phase I. Let C k be a olor class in ( H, C ) which is not a Grundy class. Then G k operates on ( H, C ) andgenerates a family of graphs of the form H S , f , m , g . In case that C k is Grundy classthen G k leaves ( H, C ) unchanged. In any case the class of vertices of color k in eachmember of G k ( H, C ) is Grundy class. Now we explain the final steps of the construction Phase II. Let (
H, C ) be anycolored graph output from Phase I. In (
H, C ) the vertex with color t + 1 is a Grundyvertex. This means that the class C t +1 is Grundy class. We operate G t on ( H, C ) andobtain a family of colored graphs. Then operate G t − on each member of this familyand obtain a new larger family denoted by G t − G t ( H, C ). Then operate G t − on themembers of the new family and continue this method by applying the operations G k , k = t − , t − , . . . ,
2. We obtain a final family of colored graphs which canbe represented by G · · · G t − G t ( H, C ). Note that each color class in each member ofthe new family is Grundy class. The family D t +1 is defined as this final family, i.e. D t +1 = (cid:91) ( H,C ):(
H,C ) is constructed in phase I G · · · G t − G t ( H, C )We call each member of D t +1 a z -atoms with z -chromatic number at least t + 1.Also for each member G ∈ D t +1 the corresponding coloring C (cid:48) of G is a z -coloringof G using t + 1 colors. This coloring is called the canonic z -coloring of G . Infact the original vertices of the initial star graph S ,t , i.e. u , u , . . . , u t +1 are color-dominating vertices and by Remark 1 the coloring C (cid:48) of G has Grundy property.Clearly, the complete graphs on one and two vertices are the only z -atoms with z -number one and two, respectively. It is also easily observed that there are two z -atoms with z -number three. They are the complete graph K and the path on 5vertices P . There are too many z -atoms with z -number four. If we generate onlytriangle-free ones then we obtain 18 such z -atoms, which are depicted in Figures 2and 3, together with a z -coloring for each of them consisting of four color-dominatingvertices. The largest one is naturally a tree and contains 14 vertices and is illustratedin Figure 2. Theorem 2.
Let t ≥ be any fixed integer. Let L be any graph and z ( L ) ≥ t + 1 .Then there exists a z -atom ( G, C ∗ ) ∈ D t +1 which is embedded in L , where C ∗ is thecanonic coloring of G . Proof.
Let C be a z -coloring of L using t + 1 colors. By Proposition 4 thereexist t + 1 vertices u , u , . . . , u t +1 of colors 1 , , . . . , t + 1 in C , respectively and acolored graph ( H, C (cid:48) ) such that for each i ≤ t , u t +1 is adjacent to u i and u i is acolor-dominating vertex of color i . Also started from the star graph consisting of thevertices u , u , . . . , u t +1 , the graph ( H, C (cid:48) ) is a subgraph of L and one of the graphsgenerated in Phase I, where the coloring C (cid:48) is the restriction of the coloring C tothe vertices of H . Denote the color classes in H by C , C , . . . , C t , C t +1 . Obviously15 t +1 = { u t +1 } is a Grundy class in H . Let D be the set of vertices of color t in H .Let w be any vertex in D . Since C has Grundy property, for each i < t there existsa neighbor of w in L whose color in C is i . Let D i be a minimal subset of verticesin L whose color in C is i and D i dominates the vertices of D . Obviously D i ispartitioned into D i ∩ V ( H ) and D i \ V ( H ). Write for simplicity ˜ D i = D i \ V ( H ).Let H (cid:48) be a (colored) subset of G induced by V ( H ) ∪ ˜ D ∪ ˜ D ∪ · · · ∪ ˜ D t − . Interpretthe vertices of ˜ D i as the vertices denoted by w i , . . . , w im i (for some suitable m i )in the operation G t . Also since D i = ˜ D i ∪ ( D i ∩ V ( H )) dominates the vertices of D ⊆ V ( H ) then there exists a subset of vertices, say tentatively Q , in D (this subset Q is in fact denoted by C it \ S i in the operation G t ) such that Q is dominated onlyby ˜ D i . Hence the vertices of Q are mapped by a surjective mapping say g i into ˜ D i (or { w i , . . . , w im i } ). Denote by C (cid:48)(cid:48) the coloring of H (cid:48) obtained by restriction of C to V ( H (cid:48) ). It follows that one of the graphs constructed in G t ( H, C (cid:48) ) is isomorphicto ( H (cid:48) , C (cid:48)(cid:48) ). By Remark 1 the class of vertices of color t in ( H (cid:48) , C (cid:48)(cid:48) ) is Grundyclass. Now we repeat the above argument for the colored graph ( H (cid:48) , C (cid:48)(cid:48) ) and thecolor t − H (cid:48) . By continuing this procedurewe obtain a colored subgraph of ( L, C ), say G , which is isomorphic to a graph inthe family G · · · G t − G t ( H, C (cid:48) ). Let C ∗ be the restriction of C to V ( G ). Now, the z -atom ( G, C ∗ ) is embedded in L , as desired. (cid:3) Theorem 2 provides a computational tool to prove upper bounds of the form z ( G ) ≤ t for z -chromatic number (and so chromatic number) of graphs, where t is anyarbitrary and fixed integer. Let G be any graph and t a fixed integer such thatno element of D t is embedded in G . Then z ( G ) ≤ t . Since D t is finite, to verifythat no element of D t is embedded in G can be done in a polynomial time steps interms of | V ( G ) | by using the following method. It is clear that the largest z -atomin D t is a smallest tree T with z ( T ) = t . It is proved in Proposition 5 that thereexists only one tree T with z ( T ) = t and with the minimum number of vertices. Itis proved that | V ( T ) | = ( t − t − + t + 2. Let f ( t ) = ( t − t − + t + 2. Let( H, C ) be an arbitrary member in D t . The graph G is our input graph. By anexhaustive search in (cid:18) | V ( G ) || V ( H ) | (cid:19) steps we can check if ( H, C ) is imbedded in G ornot. Since | V ( H ) | ≤ f ( t ) then each element of D t can be checked in | V ( G ) | f ( t ) steps.Let |D t | = g ( t ). Therefore to verify that no element of D t is embedded in G needsoverall g ( t ) | V ( G ) | f ( t ) time steps. Since t is fixed then it is polynomial in terms of | V ( G ) | .The following result uses the above-mentioned technique but we have to omit someof its lengthy and tedious details. Theorem 3.
Let G be ( K , P ) -free graph. Then z ( G ) ≤ . Proof.
Assume on the contrary that G admits a z -coloring using 4 colors. Since G is triangle-free then there exists at least one z -atom ( H, C ) from the graphs depicted16igure 2: The nine z -atoms from the whole eighteen triangle-free z -atoms with z -number four. For each one a z -coloring using four colors is illustrated, where thecolor-dominating vertices are indicated by circles.17igure 3: The remaining nine z -atoms from the whole eighteen triangle-free z -atomswith z -number four. For each one a z -coloring using four colors is illustrated, wherethe color-dominating vertices are indicated by circles.18n Figures 2 and 3 such that ( H, C ) is embedded in G , where C is the z -coloringof H using four colors illustrated in the figures. Applying the following argumentfor each of these 18 z -atoms leads to contradiction. Note that each graph ( H, C )in the figures have many induced P . But ( H, C ) is embedded in G and G is P -free. Therefore many extra edges should be added to each z -atom ( H, C ) in orderto destroy all existing induced P . But we are not allowed to add edges betweenvertices with a same color in ( H, C ). When we add extra edges to (
H, C ) satisfyingthis condition then some new induced P is created in the graph. Hence more extraedges are required to be added to the graph. Eventually, this procedure makes atriangle in the graph. This contradicts the fact that G is triangle-free. (cid:3) In this section we first determine the following quantity a k = min {| V ( T ) | : T is a tree with z ( T ) = k } . We also show that there exists a unique tree R k such that z ( R k ) = k and | V ( R k ) | = a k . Then in the light of Theorem 2 it follows that for every tree T , z ( T ) ≥ k if andonly if T contains a subtree isomorphic to R k . In this situation we can apply thefollowing result of [15]. Given any two trees T and R , to determine whether R isisomorphic to a subtree of T can be solved in polynomial time.In the following we try to construct a graph T with smallest possible number ofvertices satisfying z ( T ) = k and subject to condition that T is acyclic. Let C be a z -coloring with k colors. There should be color-dominating vertices u , . . . , u k suchthat u j is of color j in C and that u k is adjacent to each u j , j (cid:54) = k . Consider u k as a root and expose the rest of vertices from top to down. Hence, u , . . . , u k − arethe children of u k and lie in the second or lower level. For each j ∈ { , . . . , k − } , u j is color-dominating vertex of color j , so it needs k − , , . . . , j − , j + 1 , . . . , k −
1. All of such neighbors are distinct because the graphto be constructed is acyclic. Place these neighbors in a third level. But the coloring C has Grundy property, hence the vertices of the third level need suitable newneighbors in the forth level and so on. We denote by R k the tree constructedaccording to this procedure. Obviously R and R are isomorphic the completegraphs K and K , respectively. Observe that R is isomorphic to the path on 5vertices P . Figure 4 depicts R and R with their corresponding labeling. The tree R k is constructed so that z ( R k ) ≥ k and is minimal tree with this property. Alsoeach vertex in the tree has at most one and k − R k ) = k − z ( R k ) = k . We have | R k | = a k by the definition of a k . Note that a = 1 , a = 2 , a = 5 , a = 14. In the followingwe prove that a k = 2 a k − + 2 k − − k , for each k ≥ b bb b
31 22 1 bbb bb b b bb bb bb b
41 2 32 3 1 3 1 22 1 11 R R Figure 4: The trees R and R with their canonic labelingsDenote by S k the subtree of R k whose root is u . In the following we make a closeconnection between S k and T k , where T k is the only tree atom whose Grundy numberis k . Consider S k and its coloring from C which is a z -coloring with k colors. Thevertex u has color 1 in this coloring. As we mentioned, u has k − , , . . . , k in C . For a moment assign color k to u and attach a leaf of color1 to each neighbor of u as well as the vertex u itself. Denote the resulting tree by˜ S k . The resulting coloring for ˜ S k is a Grundy coloring for ˜ S k with k colors. Since R k is minimal then ˜ S k is minimal with respect to Grundy coloring. In other words,˜ S k is isomorphic to T k . In order to obtain a precise connection, we do the following.In the Grundy coloring of T k with k colors there exists exactly one vertex of color k . Denote this vertex by the very notation u . The vertex u (in T k ) has k − , , . . . , k −
1. Denote these neighbors by w , . . . , w k − . For each i ≥ w i has a neighbor say p i of color 1 in T k . Now remove u , w , p , . . . , p k − from T k . Denote the resulting graph by L k . The above argument shows that L k and S k \ { u } are isomorphic and the label of each vertex v in L k is the same as theisomorphic copy of v in S k \ { u } . Note that | V ( L k ) | = | V ( T k ) | − k . It follows that | V ( S k \ { u } ) | = 2 k − − k .Consider now U k = R k \ ( S k \ { u } ). Each vertex in U k has a neighbor of color1. See this situation in Figure 4. Remove from U k these neighbors of color 1 andthen decrease the color of each vertex by one. This results in a z -coloring with k − | V ( U k ) | / R k − that the very remaining tree is isomorphic to R k − . Therefore | V ( U k ) | = 2 | V ( R k − ) | . Finally, we obtain that | V ( R k ) | = 2 | V ( R k − ) | + 2 k − − k , asdesired.In order to determine the exact value of | V ( R k ) | = a k we apply the generatingfunction method. Set f ( x ) = (cid:88) ∞ k =1 a k x k . We have a k = 2 a k − + 2 k − − k for each k ≥ a = 0 , a = 1. We have, 20 (cid:88) k =2 a k x k = 2 ∞ (cid:88) k =2 a k − x k + ∞ (cid:88) k =2 k − x k − ∞ (cid:88) k =2 kx k . By computations and power series expansions we obtain f ( x ) = 2 x (1 − x ) − x (1 − x ) (1 − x ) + 2 x (1 − x ) . By solving f ( x ) in terms of power series, we obtain a k = ( k − k − + k + 2. Thefollowing proposition and corollary are immediate. Proposition 5.
Let T be any tree and z ( T ) = k for some integer k . Then | V ( T ) | ≥ ( k − k − + k + 2 . Moreover, inequality holds only for the tree R k . Remark 2.
The z -chromatic number of trees is determined by a polynomial timealgorithm. In the following for every graph H denote by ˜ H the graph obtained by attaching aleaf to each vertex of H . The following result will be used in the next theorem. Theorem 4.
For any tree T , Γ( T ) ≤ z ( T ) . Proof.
Take the trees R k − and T k − and consider the trees ˜ R k − and ˜ T k − . Let u and w be the roots (with color k −
1) of R k − and T k − , respectively. Now connect˜ R k − and ˜ T k − by adding an edge between u and w and obtain a new tree R . By theexplanation concerning the construction of R k , note that R k is a subtree of R . In factthe color of u and w will be respectively k and 1, in the canonic coloring of R k . Toprove the theorem it is enough to prove that the tree T with Γ( T ) ≥ k contains R as subgraph. Consider a Grundy coloring C of T using k color classes C , . . . , C k .We can embed ˜ T k − in C ∪ · · · ∪ C k and (by induction on k ) embed R k − in the next( k − color classes of C . Hence ˜ R k − is embedded in C ∪ C k +1 ∪ · · · ∪ C ( k − + k .Since the whole coloring is Grundy then the embedding can be done so that u isadjacent to w . Note that ( k − + k ≤ k . This completes the proof. (cid:3) Recall that T k is the unique smallest tree with Γ( T k ) = k . The following resultcompares z ( T k ) and Γ( T k ). Theorem 5. lim k →∞ (Γ( T k ) − z ( T k )) = + ∞ . roof. Set for simplicity z ( T k ) = p k . By Proposition 5 we have2 k − = | T k | ≥ ( p k − p k − + p k + 2 > ( p k − p k − . It follows that 2 k − p k > p k − k − p k > log( p k − k − p k (cid:57) ∞ thenthere exits a number N such that for each k , k − p k ≤ N . Hence log( p k − < N and p k < N + 3 for each k . But by Theorem 4, √ k ≤ p k < N + 3, a contradiction. (cid:3) Since the z -coloring heuristic and z -chromatic number are newly defined chromaticconcepts, many chromatic and algorithmic problems can be raised for them. Is z -chromatic number of graphs an N P -complete parameter? As a computationalproject, it is useful to produce the bank of z -atoms, at least for low z -numbers.Another research area is to prove upper bound results such as Theorem 3, usingthe bank of z -atoms. For example, what is the best possible upper bound for the z -chromatic number of ( K , P )-free graphs? We believe that a competitive coloringheuristic, which we denote by IZ , is obtained from the z -coloring heuristic. Assumethat we applied the z -coloring heuristic for a graph G and obtained the color classes C , C , . . . , C k . We can repeat the heuristic in the reverse order i.e. from C to C k . Also, let σ be any random permutation of { , , . . . , k } . We can repeat theheuristic by scanning the classes according to the order C σ (1) , C σ (2) , . . . , C σ ( k ) . Callthis iterated form of the heuristic, iterated z -coloring heuristic (shortly IZ ). Sinceevery z -coloring has Grundy property then IZ is better than the iterated greedy IG heuristic of Culberson [5, 6]. The author thank anonymous referees for their useful comments.
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