Equitable Colorings of Corona Multiproducts of Graphs
EEquitable Colorings of CoronaMultiproducts of Graphs ∗ Hanna Furma´nczyk † , Marek Kubale ‡ Vahan V. Mkrtchyan § Abstract
A graph is equitably k -colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two setsdiffer by at most one. The smallest k for which such a coloring exists is knownas the equitable chromatic number of G and denoted χ = ( G ). It is known thatthis problem is NP-hard in general case and remains so for corona graphs. In[12] Lin i Chang studied equitable coloring of Cartesian products of graphs. Inthis paper we consider the same model of coloring in the case of corona prod-ucts of graphs. In particular, we obtain some results regarding the equitablechromatic number for l -corona product G ◦ l H , where G is an equitably 3- or4-colorable graph and H is an r -partite graph, a path, a cycle or a completegraph. Our proofs are constructive in that they lead to polynomial algorithmsfor equitable coloring of such graph products provided that there is given anequitable coloring of G . Moreover, we confirm Equitable Coloring Conjecturefor corona products of such graphs. This paper extends our results from [8]. Keywords: corona graph, equitable chromatic number, equitable coloring conjec-ture, equitable graph coloring, NP-completeness, polynomial algorithm, multiproductof graphs
All graphs considered in this paper are finite, connected and simple, i.e. undirected,loopless and without multiple edges.If the set of vertices of a graph G can be partitioned into k (possibly empty) classes V , V , ...., V k such that each V i is an independent set and the condition || V i | − | V j || ≤ ∗ Project has been partially supported by Narodowe Centrum Nauki under contract DEC-2011/02/A/ST6/00201 † Institute of Informatics, University of Gda´nsk, Wita Stwosza 57, 80-952 Gda´nsk, Poland.e-mail: [email protected] ‡ Department of Algorithms and System Modelling, Technical University of Gda´nsk, Narutowicza11/12, 80-233 Gda´nsk, Poland. e-mail: [email protected] § Department of Informatics and Applied Mathematics, Yerevan State University, Armenia. e-mail: [email protected] a r X i v : . [ c s . D M ] O c t quitable Colorings of Corona Multiproducts of Graphs i, j ), then G is said to be equitably k-colorable . In the case, whereeach color is used the same number of times, i.e. | V i | = | V j | for every pair ( i, j ), graph G is said to be strongly equitably k -colorable . The smallest integer k for which G isequitably k -colorable is known as the equitable chromatic number of G and denoted χ = ( G ) [13]. Since equitable coloring is a proper coloring with additional condition,the inequality χ ( G ) ≤ χ = ( G ) holds for any graph G .In some discrete industrial systems we can encounter the problem of equitablepartitioning of a system with binary conflict relations into conflict-free subsystems.Such situations can be modeled by means of equitable graph coloring. For example, inthe garbage collection problem the vertices of the graph represent garbage collectionroutes and a pair of vertices is joined by an edge if the corresponding routes shouldnot be run on the same day. The problem of assigning one of the six days of thework week to each route thus reduces to the problem of 6-coloring of the graph [13].In practice it might be desirable to have an approximately equal number of routesrun on each of the six days. So we have to color the graph in an equitable way withsix colors. Other applications of equitable coloring can be found in scheduling andtimetabling.The notion of equitable colorability was introduced by Meyer [13]. However, anearlier work of Hajnal and Szemer´edi [9] showed that a graph G with maximal degree∆ is equitably k -colorable if k ≥ ∆ + 1. Recently, Kierstead et al. [10] have givenan O (∆ | V ( G ) | )-time algorithm for equitable (∆ + 1)-coloring of graph G . In 1973,Meyer [13] formulated the following conjecture: Conjecture 1 (Equitable Coloring Conjecture (ECC)) . For any connected graph G other than a complete graph or an odd cycle, χ = ( G ) ≤ ∆ . This conjecture has been verified for all graphs on six or fewer vertices. Lih andWu [11] proved that the Equitable Coloring Conjecture is true for all bipartite graphs.Wang and Zhang [14] considered a broader class of graphs, namely r -partite graphs.They proved that Meyer’s conjecture is true for complete graphs from this class. Also,the conjecture (or even the stronger one) was confirmed for outerplanar graphs [15]and planar graphs with maximum degree at least 13 [16].The corona of two graphs G and H is a graph G ◦ H formed from one copy of G and | V ( G ) | copies of H where the i th vertex of G is adjacent to every vertex inthe i th copy of H . For any integer l ≥
2, we define the graph G ◦ l H recursivelyfrom G ◦ H as G ◦ l H = ( G ◦ l − H ) ◦ H (cf. Fig. 1). Graph G ◦ l H is also named as l -corona product of G and H . Such type of graph product was introduced by Fruchtand Harary in 1970 [5].A straightforward reduction from graph coloring to equitable coloring by addingsufficiently many isolated vertices to a graph, proves that it is NP-complete to testwhether a graph has an equitable coloring with a given number of colors (greaterthan two). Furma´nczyk et al. [8] proved that the problem remains NP-complete forcorona graphs. Bodlaender and Fomin [1] showed that equitable coloring problem canbe solved to optimality in polynomial time for trees (previously known due to Chenand Lih [3]) and outerplanar graphs. A polynomial time algorithm is also known forequitable coloring of split graphs [2], cubic graphs [4] and some coronas [8]. quitable Colorings of Corona Multiproducts of Graphs l -corona product of graphswhile in Section 3 we give some results concerning the equitable colorability of l -corona products of some graphs and r -partite graphs. Next, in Section 4 we consider l -corona products of graphs G with χ = ( G ) ≤ l -corona products of these graphs G and paths. In this way we extend the classof graphs that can be colored optimally in polynomial time and confirm the ECCconjecture [8].Figure 1: Example of graphs: a) C ; b) C ◦ K ; c) C ◦ K . quitable Colorings of Corona Multiproducts of Graphs It is known that χ = ( G ◦ K m ) = m + 1 for every graph G such that χ ( G ) ≤ m + 1[8]. Since the obtained graph G ◦ K m is ( m + 1)-colorable, the graph G ◦ K m is alsoequitably ( m + 1)-colorable, and so on. This result can be easy generalized to the l -corona product. Proposition 2.1. If G is a graph with χ ( G ) ≤ m + 1 , then χ = ( G ◦ l K m ) = m + 1 ,for any l ≥ . Let us note that since G is connected, the maximum degree of the corona ∆( G ◦ l K m ) is equal to ∆( G ) + m · l . Since m + 1 ≤ ∆( G ) + m · l , so ECC is true for suchgraphs.Let us also notice that we immediately get an upper bound on the equitablechromatic number: χ = ( G ◦ l H ) ≤ m + 1 , where l ≥ χ ( G ) ≤ m + 1 and graph H is of order m . r -partitegraphs In this section we consider corona products of graph G and r -partite graphs, where G fulfills some additional conditions. Theorem 3.1.
Let G be an equitably k -colorable graph on n vertices and let H be a ( k − -partite graph. If k | n , then for any l ≥ χ = ( G ◦ l H ) ≤ k. Proof.
The proof is by induction on l .Step 1: For l = 1 the theorem holds due to the following.Suppose V ( G ) = V ∪ V ∪ · · · ∪ V k , where V , . . . , V k are independent sets eachof size n/k . This means that they form an equitable k -coloring of G . For eachvertex z ∈ V ( G ), let H z = ( X z , . . . , X zk − , E z ) be the copy of ( k − H = ( X , . . . , X k − , E ) in G ◦ H corresponding to z . Let V (cid:48) = V ∪ (cid:83) z ∈ V X z ∪ · · · ∪ (cid:83) z ∈ V k X zk − ,V (cid:48) = V ∪ (cid:83) z ∈ V X z ∪ · · · ∪ (cid:83) z ∈ V k X zk − ∪ (cid:83) z ∈ V X zk − , ... V (cid:48) k − = V k − ∪ (cid:83) z ∈ V k X z ∪ (cid:83) z ∈ V X z ∪ · · · ∪ (cid:83) z ∈ V k − X zk − ,V (cid:48) k = V k ∪ (cid:83) z ∈ V X z ∪ · · · ∪ (cid:83) z ∈ V k − X zk − . quitable Colorings of Corona Multiproducts of Graphs V ( G ◦ H ) = V (cid:48) ∪ · · · ∪ V (cid:48) k is an equitable k -coloring of G ◦ H .In this coloring each of k colors is used exactly n (1+ | X | + · · · + | X k − | ) /k times.Step 2: Suppose Theorem 3.1 holds for some l ≥ χ = (( G ◦ l H ) ◦ H ) ≤ k. Let us note that if k | n thenthe cardinality of vertex set of G ◦ l H , which is equal to n ( | V ( H ) | + 1) l , is alsodivisable by k . So using the inductive hypothesis we get immediately the thesis.Let us note that the bound on the equitable chromatic number given in Theorem3.1 holds for corona multiproducts of equitably k -colorable graph G on n vertices and r -partite graph H where r ≤ k − k | n . Let assume that G fulfills the assumptionof Theorem 3.1 and graph H is r -partite where r < k −
1. We can add extra edges tograph H , without adding new vertices, until a new graph H (cid:48) is ( k − G ◦ l H (cid:48) is equitably k -colorable. Since G ◦ l H is a subgraphof G ◦ l H (cid:48) , it is also equitably k -colorable. Corollary 3.2.
Let G be an equitably k -colorable graph on n vertices and let H bean r -partite graph where r ≤ k − . If k | n , then for any l ≥ χ = ( G ◦ l H ) ≤ k. We will consider two cases: the first one for even cycles, and the second one for oddcycles.
Theorem 4.1.
Let G be an equitably 3-colorable graph on n ≥ vertices and let k ≥ , l ≥ . If | n or k = 2 , then χ = ( G ◦ l C k ) = 3 . Proof.
The first part of the theorem, for 3 | n , follows from Theorem 3.1. Of course,we cannot use fewer than three colors.The case when k = 2 was partially considered in [8]. The authors proved that if G is an equitably 3-colorable graph on n ≥ χ = ( G ◦ C ) = 3. This meansthat our theorem is true for l = 1. The farther part of this proof is by induction onthe number l , similarily as it was in the proof of Theorem 3.1.We also know that in the remaining cases, i.e. when G is equitably 4-colorable or3 (cid:45) n , we need more than three colors for equitable coloring of G ◦ C k [8]. quitable Colorings of Corona Multiproducts of Graphs Theorem 4.2.
Let G be an equitably 4-colorable graph on n ≥ vertices and let k ≥ , l ≥ . Then χ = ( G ◦ l C k ) ≤ . Proof.
Let us consider two cases.Case 1: 3 | n We consider two subcases, depending on n mod 4.Subcase 1.1: n mod 4 = 0.The thesis follows immediately from Corollary 3.2.Subcase 1.2: n mod 4 (cid:54) = 0.First, we will show that our theorem is true for l = 1 and then by inductionon l we will get the thesis for multicoronas G ◦ l C k , l ≥ G has been colored equitably with 4 colors and the cardi-nalities of color classes are arranged in non increasing way. We order thevertices of G : v , v , . . . , v n in such a way that vertex v i is colored withcolor i mod 4, and we use color 4 instead of 0.We color first 4 x copies of C k using in the i th copy k times color ( i mod 4+1) mod 4, (cid:100) k/ (cid:101) times color ( i mod 4 + 2) mod 4 and (cid:98) k/ (cid:99) times color( i mod 4 + 3) mod 4, where x is defined below. In this part each color isused the same number of times.(i) n mod 4 = 1.Since n is a multiple of three, there is p ≥
0, such that n = 12 p + 9. Inthis subcase x is defined as 4 p + 1. Finally, we have to color verticesin last five copies of C k in corona G ◦ C k as follows: • we color the 1st copy using k times color 2, (cid:100) k/ (cid:101) times color 3, (cid:98) k/ (cid:99) times color 4, • we color the 2nd copy using k times color 1, (cid:100) k/ (cid:101) times color 4, (cid:98) k/ (cid:99) times color 3, • we color the 3rd copy using k times color 1, k times color 4, • we color the 4th copy using k times color 2, (cid:100) k/ (cid:101) times color 3, (cid:98) k/ (cid:99) times color 1, • we color the 5th copy using k times color 3, (cid:100) k/ (cid:101) times color 4, (cid:98) k/ (cid:99) times color 2.(ii) n mod 4 = 2.Since n is a multiple of three, there is p ≥
0, such that n = 12 p + 6.In this subcase x is also defined as 4 p + 1. Finally, we have to colorvertices in last two copies of C k in corona G ◦ C k as follows: • we color the 1st copy using k times color 2, (cid:100) k/ (cid:101) times color 3, (cid:98) k/ (cid:99) times color 4, quitable Colorings of Corona Multiproducts of Graphs • we color the 2nd copy using k times color 1, (cid:100) k/ (cid:101) times color 4, (cid:98) k/ (cid:99) times color 3.(iii) n mod 4 = 3.Since n is a multiple of three, there is p ≥
0, such that n = 12 p + 3. Inthis subcase x is defined as 4 p . Finally, we have to color the verticesin the last three copies of C k in corona G ◦ C k as follows: • we color the 1st copy using k times color 3, (cid:100) k/ (cid:101) times color 4, (cid:98) k/ (cid:99) times color 2, • we color the 2nd copy using k times color 1, (cid:100) k/ (cid:101) times color 4, (cid:98) k/ (cid:99) times color 3, • we color the 3rd copy using k times color 2, (cid:100) k/ (cid:101) times color 4, (cid:98) k/ (cid:99) times color 1.It can be easily checked that each of the above colorings, in all subcases,is an equitable 4-coloring of G ◦ C k .Case 2: 3 (cid:45) n .It follows from [8] that if G is an equitably 4-colorable graph on n vertices, n ≥ (cid:45) n , then χ = ( G ◦ C k ) = 4 for k ≥
3. This means that our theoremis true for l = 1. Therefore, by induction on l , we get the desired result.It turns out that in the case when the number of vertices of graph G is not divisibleby three, the weak inequality becomes equality. Theorem 4.3.
Let G be an equitably 4-colorable graph on n ≥ vertices and let k ≥ , l ≥ . If (cid:45) n , then χ = ( G ◦ l C k ) = 4 . Proof.
All we need is the proof that we cannot use fewer colors.If χ ( G ) = 4 then of course χ = ( G ◦ l C k ) >
3, for any l . Let us assume that χ ( G ) ≤
3. Note that any 3-coloring of G uniquely determines a 3-coloring of G ◦ l C k , l ≥
2. When we color the vertices in a copy of C k linked to a vertex of G ◦ l − C k ,we use two available colors. It is not hard to notice that the difference betweencardinalities of color clases is the smallest when 3-coloring of G is strongly equitable.In our case, when n is not divisible by three, a strongly equitable coloring does notexist. If the difference between cardinalities of every two color classes of G is greaterthan or equal to one, any 3-coloring of G ◦ l C k cannot be equitable. This followsfrom the following reasoning.We claim that every equitable (not strongly) 3-coloring of G determines 3-coloringof G ◦ l C k with maximum difference among the color classes equaling to ( k − l , l ≥ l . quitable Colorings of Corona Multiproducts of Graphs l = 1.(i) n mod 3 = 1Cardinalities of color classes for colors 1, 2 and 3 are equal to (cid:98) n/ (cid:99) (2 k +1) + 1, (cid:98) n/ (cid:99) (2 k + 1) + k and (cid:98) n/ (cid:99) (2 k + 1) + k , respectively. The maximumdifference between color classes is equal to ( k − . Our claim holds.(ii) n mod 3 = 2Cardinalities of color classes for colors 1, 2 and 3 are equal to (cid:98) n/ (cid:99) (2 k +1) + 1 + k , (cid:98) n/ (cid:99) (2 k + 1) + 1 + k and (cid:98) n/ (cid:99) (2 k + 1) + 2 k , respectively. Themaximum difference between color classes is also equal to ( k − . So, ourclaim holds also in this case.Step 2: Induction hypothesis for l ≥ . We assume that maximum difference between color classes in 3-coloring of G ◦ l C k is equal to ( k − l .Step 3: The proof that the difference for l + 1 does not exceed ( k − l +1 . It is easy to see that we have to compute the difference between cardinalities ofcolor class for color 3 ( | C l +13 | ) and color 1 ( | C l +11 | ). Again, we have to considertwo subcases:(i) n mod 3 = 1(a) | C l | = x for some x , and | C l | = | C l | = x + ( k − l .Let us notice that | C l +11 | = | C l | +( | C l | + | C l | ) · k = x +2 xk +2( k − l k -we can color only these vertices of copies of C k in G ◦ l +1 C k with colorone that are not adjacent to vertex colored with one in G ◦ l C k , while | C l +13 | = x + ( k − l + (2 x + ( k − l ) k . The difference | C l +13 | − | C l +11 | =( k − l +1 .(b) | C l | = x for some x , and | C l | = | C l | = x − ( k − l .Analogously to above.(ii) n mod 3 = 2(a) | C l | = | C l | = x for some x , and | C l | = x + ( k − l .Let us notice that | C l +11 | = | C l | + ( | C l | + | C l | ) · k = x + 2 xk + ( k − l k -we can color only these vertices of copies of C k in G ◦ l +1 C k with colorone that are not adjacent to vertex colored with one in G ◦ l C k , while | C l +13 | = x + ( k − l + 2 xk . The difference | C l +11 | − | C l +13 | = ( k − l +1 .(b) | C l | = | C l | = x for some x , and | C l | = x − ( k − l .Analogously to above.Summing up, even when 3 | (2 k +1) and G is equitably (not strongly) 3-colorable,corona G ◦ l C k , l ≥ quitable Colorings of Corona Multiproducts of Graphs Theorem 4.4.
Let G be an equitably 4-colorable graph on n ≥ vertices and let k ≥ . Then for any l ≥ we have χ = (cid:0) G ◦ l C k +1 (cid:1) = 4 . Proof.
The inequality χ = (cid:0) G ◦ l C k +1 (cid:1) ≤ K ◦ C k +1 is a subgraph of G ◦ l C k +1 , we cannotuse fewer colors.We have considered equitable coloring of corona product of graphs on at least twovertices and cycles. Now, for the sake of completeness, we consider equitable coloringsof corona products of one isolated vertex and cycles. We have noticed in [8] that χ = ( K ◦ C m ) = (cid:26) , if m = 3 , (cid:6) m (cid:7) + 1 , if m > . (1)The value of equitable chromatic number of multicorona K ◦ l C m can be arbitrarilylarge for l = 1 (cf. Equality (1)). The situation changes significantly for larger valuesof l . Theorem 4.5.
Let m ≥ and l ≥ . Then χ = ( K ◦ l C m ) = (cid:26) , if m = 4 , , otherwise.Proof. Let us consider three cases.Case 1: m = 3Since C = K , our thesis follows immediately from Proposition 2.1.Case 2: m = 4By Equality (1) χ = ( K ◦ C ) = 3. For l ≥ m ≥ l .Step 1: For l = 2 the theorem holds due to the following. We consider two casesdepending on the parity of m . quitable Colorings of Corona Multiproducts of Graphs m is even.First, we prove that χ = ( K ◦ C k ) >
3. Let us notice that 3-coloringof K ◦ C k is unique up to permutation of colors. The vertex of K is assigned, say, color 1, vertices of C k in K ◦ C k adjacent to vertexcolored 1, must be colored with 2 and 3, alternately. Next, we assigntwo available colors to vertices in 2 k + 1 copies of C k in K ◦ C k .Cardinalities of color classes in such a 3-coloring are equal to 1 + 2 k · k and twice k + ( k + 1) · k , respectively. It is easy to see that this coloringis not equitable for k ≥ K ◦ C k .The cardinalities of color classes in every such coloring are equal to (cid:100)| V ( K ◦ C k ) | / (cid:101) = (cid:100) (2 k + 1)(2 k + 1) / (cid:101) , (cid:100) [(2 k + 1)(2 k + 1) − / (cid:101) , (cid:100) [(2 k + 1)(2 k + 1) − / (cid:101) and (cid:100) [(2 k + 1)(2 k + 1) − / (cid:101) , respectively.First, we color the vertex of K with color 1, next the vertices of C k in K ◦ C k with colors 2, 3 and 4 using them (cid:100) k/ (cid:101) , (cid:100) (2 k − / (cid:101) and (cid:100) (2 k − / (cid:101) times, respectively. Finally, we color vertices in 2 k + 1copies of C k using each time three allowed colors. In each copy everyallowed color is used (cid:100) k/ (cid:101) or (cid:98) k/ (cid:99) times. One can verify that suchequitable 4-coloring of K ◦ C k exists for each k ≥ m is odd.Let us notice that | V ( K ◦ C k +1 ) | = (2 k +2)(2 k +2) = 4( k +1) . Thismeans that each of four colors must be used exactly ( k + 1) times inevery equitable coloring.Graph K ◦ C k +1 consists of 2 k + 2 copies of C k +1 joined to verticesof K ◦ C k +1 appropriately. The equitable 4-coloring of K ◦ C k +1 isformed as follows: • the vertex of K is colored with 1 • the remaining vertices of K ◦ C k +1 are assigned colors 2, 3 and 4with cardinalities equal to k , k and 1, respectively • the copy of C k +1 in K ◦ C k +1 joined to vertex colored 1 isassigned colors 2, 3 and 4 with cardinalities equal to 1, k and k ,respectively • copies of C k +1 in K ◦ C k +1 joined to vertex colored 2 are assignedcolors 1, 3 and 4 with cardinalities in each cycle equal to 1, k and k , respectively • copies of C k +1 in K ◦ C k +1 joined to vertex colored 3 are assignedcolors 1, 2 and 4 with cardinalities in each cycle equal to k , k and1, respectively • copies of C k +1 in K ◦ C k +1 joined to vertex colored 4 are assignedcolors 1, 2 and 3 with cardinalities in each cycle C k +1 equal to k , k and 1, respectivelyIn such a coloring every color is used exactly ( k + 1) times. Let usnotice that we cannot use fewer colors, because each graph K ◦ C k +1 quitable Colorings of Corona Multiproducts of Graphs K . Hence χ = ( K ◦ C k +1 ) = 4.Step 2: Induction hypothesis.
Suppose Theorem 4.5 holds for some l ≥ χ = (( K ◦ l C m ) ◦ C m ) = 4.We consider two cases.(i) m is even.The inequality χ = ( K ◦ l C m ) ≤ χ = ( K ◦ l C k ) ≤ k ≥ l ≥ χ ( K ◦ l C k ) = 3 (graph K ◦ l C k contains K ), then we onlyneed to fix equitable 3-coloring. Let us notice that every 3-coloring of K ◦ l C k is unique up to permutation of colors. We again claim thatin such a 3-coloring the difference between color classes is equal to( k − l . The proof of our claim is analogous to the proof of Theorem4.3.It follows that such a coloring is not equitable for k ≥
3, a contradic-tion.(ii) m is odd.The thesis follows from Theorem 4.4. Since graph G ◦ l P m , m ≥ G ◦ l C m , and P m is bipartite, Theorems 3.1 and 4.1 imply Corollary 5.1.
Let G be an equitably 3-colorable graph on n vertices, and let m, n, l ≥ . If m = 4 or | n , then χ = ( G ◦ l P m ) = 3 . It turns out that there are more graphs (corona multiproduct with paths) thatcan be equitably colored with three colors.
Theorem 5.2.
Let G be an equitably 3-colorable graph on n ≥ vertices and let l ≥ . If m = 2 , , , then χ = ( G ◦ l P m ) = 3 . quitable Colorings of Corona Multiproducts of Graphs Proof.
The authors proved [8] that if G is an equitably 3-colorable graph on n ≥ χ = ( G ◦ P m ) = 3 for m = 2 , ,
5. This means that our theorem holds for l = 1. The farther part of the proof is by induction on the number l .In the remaining cases of m we sometimes have to use four colors.Since G ◦ l P m , is a subgraph of G ◦ l C m , using Theorem 4.2 we get the following Corollary 5.3.
Let G be an equitably 4-colorable graph on n ≥ vertices and let l ≥ , m ≥ . Then χ = (cid:0) G ◦ l P m (cid:1) ≤ . Now, we will consider equitable coloring of corona of K and paths. Since K ◦ P m is a fan, we have: χ = ( K ◦ P m ) = (cid:108) m (cid:109) + 1for m ≥ Corollary 5.4.
Let m, l ≥ . Then χ = ( K ◦ l P m ) (cid:26) = 3 , if m = 4 ≤ , otherwise. We can precise this result as follows.
Theorem 5.5.
Let m, l ≥ . Then χ = ( K ◦ l P m ) = 3 , if m = 2 , , ,
5= 4 , if m ≥ and m even ≤ , otherwise.Proof. Case 1: m = 2 , , , l = 2.(i) m = 2Any 3-coloring of K ◦ P leads to cardinalities of all color classes equalto 3.(ii) m = 3We are able to color K ◦ P with three colors in such a way that cardi-nalities of color classes are equal to 6, 5 and 5, respectively.(iii) m = 4This case was considered in Corollary 5.4.(iv) m = 5We are able to color K ◦ P with three colors in such a way that cardi-nalities of all color classes are equal to 12. quitable Colorings of Corona Multiproducts of Graphs l ≥ m ≥ and m is even. According to Corollary 5.3 we only have to prove that we cannot use a smallernumber of colors. The argument is the same as in the proof of Theorem 4.5.
In the paper we have given some results concerning l -corona products that confirmEquitable Coloring Conjecture. It turns out that the ECC conjecture follows forevery l -corona product G ◦ l H , where graph H is on m vertices and graph G can beproperly colored with m − l -corona products G ◦ l H that can be colored with 3 or 4 colors efficiently. Themain of our results are summarized in Table 1. (cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104)(cid:104) G H bipartite even cycles C k odd paths P k graphs k = 2 k ≥ ≤ k ≤ k ≥ | n G on n ≥ (cid:45) n | n ≤ ≤ ≤ ≤ ≤ G on n ≥ (cid:45) n G ◦ l H .Of course, the complexity of equitable coloring of G ◦ l H depends on the complexityof equitable 3- or 4-coloring of graph G , which is generally NP-hard. More precisely,since the time spend to color any vertex of H is constant, such a coloring of graphsunder consideration can be done in time O ( g ( n ) · n (cid:48) ), where g ( n ) is the complexity ofequitable 3- or 4-coloring of n -vertex graph G and n (cid:48) is the number of vertices in theremaining part of G ◦ l H . However, the following graphs: • broken spoke wheels [6], • reels [6], • cubic graphs except K [4], • some graph products [7, 12]admit equitable 3-coloring in polynomial time, and so do the corresponding multi-coronas. quitable Colorings of Corona Multiproducts of Graphs References [1] H.L. Bodleander, F.V. Fomin, Equitable colorings of bounded treewidth graphs,
Theor. Comput. Sci. m -bounded coloring of splitgraphs, in: Combinatorics and Computer Science (Brest, 1995) LCNS 1120,Springer (1996), 1–5.[3] B.L. Chen and K.W. Lih, Equitable coloring of trees,
J. Combin. Theory Ser. B
61 (1994), 83–87.[4] B.L. Chen, K.W. Lih and P.L. Wu, Equitable coloring and the maximum degree,
Europ. J. Combinatorics
15 (1994), 443–447.[5] R. Frucht, F. Harary, On the corona of two graphs,
Aequationes Math.
Graph Colorings , (M. Kubale,ed.) American Mathematical Society Providence, Rhode Island (2004).[7] H. Furma´nczyk, Equitable coloring of graph products,
Opuscula Mathematica
Equitable coloring of coronaproducts of graphs , submitted.[9] A. Hajnal, E. Szemeredi, Proof of a conjecture of Erd¨os, in:
Combinatorial Theoryand Its Applications, II
Combinatorica
Disc. Math.
Disc.App. Math.
160 (2012), 239–247.[13] W. Meyer, Equitable coloring,
Amer. Math. Monthly
80 (1973), 920–922.[14] W. Wang, K. Zhang, Equitable colorings of line graphs and complete r -partitegraphs, Systems Science and Mathematical Sciences
13 (2000), 190–194.[15] H.P. Yap, Y. Zhang, The Equitable ∆-Coloring Conjecture holds for outerplanargraphs,
Bulletin of the Inst. of Math. Academia Sinica
25 (1997), 143–149.[16] H.P. Yap, Y. Zhang, Equitable colourings of planar graphs,