aa r X i v : . [ m a t h . C O ] M a r On Colorings of Graph Powers
Hossein Hajiabolhassan
Department of Mathematical SciencesShahid Beheshti UniversityP.O. Box , Tehran, Iran [email protected]
Abstract
In this paper, some results concerning the colorings of graph powers are pre-sented. The notion of helical graphs is introduced. We show that such graphsare hom-universal with respect to high odd-girth graphs whose (2 t + 1)st poweris bounded by a Kneser graph. Also, we consider the problem of existence ofhomomorphism to odd cycles. We prove that such homomorphism to a (2 k +1)-cycle exists if and only if the chromatic number of the (2 k + 1)st power of S ( G )is less than or equal to 3, where S ( G ) is the 2-subdivision of G . We also con-sider Neˇsetˇril’s Pentagon problem. This problem is about the existence of highgirth cubic graphs which are not homomorphic to the cycle of size five. Severalproblems which are closely related to Neˇsetˇril’s problem are introduced andtheir relations are presented. Keywords: graph homomorphism, graph coloring, circular coloring.
Subject classification: 05C
Throughout this paper we only consider finite graphs. A homomorphism f : G −→ H from a graph G to a graph H is a map f : V ( G ) −→ V ( H ) such that uv ∈ E ( G )implies f ( u ) f ( v ) ∈ E ( H ). The existence of a homomorphism is indicated by thesymbol G −→ H . Two graphs G and H are homomorphically equivalent if G −→ H and H −→ G . Also, the symbol Hom( G, H ) is used to denote the set of allhomomorphisms from G to H (for more on graph homomorphisms see [2, 3, 8, 13]).If n and d are positive integers with n ≥ d , then the circular complete graph K ( n,d ) is the graph with vertex set { v , v , . . . , v n − } in which v i is connected to v j if andonly if d ≤ | i − j | ≤ n − d . A graph G is said to be ( n, d )-colorable if G admits ahomomorphism to K ( n,d ) . The circular chromatic number (also known as the starchromatic number [31]) χ c ( G ) of a graph G is the minimum of those ratios nd forwhich gcd ( n, d ) = 1 and such that G admits a homomorphism to K ( n,d ) . It can beshown that one may only consider onto-vertex homomorphisms [33]. We denote by[ m ] the set { , , . . . , m } , and denote by (cid:0) [ m ] n (cid:1) the collection of all n -subsets of [ m ].For a given subset A ⊆ [ m ], the complement of A in [ m ] is denoted by A . The Kneser graph KG ( m, n ) is the graph with vertex set (cid:0) [ m ] n (cid:1) , in which A is connectedto B if and only if A ∩ B = ∅ . It was conjectured by Kneser [16] in 1955, and provedby Lov´asz [20] in 1978, that χ ( KG ( m, n )) = m − n + 2. A subset S of [ m ] is called2- stable if 2 ≤ | x − y | ≤ m − x and y of S . The Schrijver This research was partially supported by Shahid Beheshti University. Correspondence should be addressed to [email protected] . raph SG ( m, n ) is the subgraph of KG ( m, n ) induced by all 2-stable n -subsets of[ m ]. It was proved by Schrijver [27] that χ ( SG ( m, n )) = χ ( KG ( m, n )) and thatevery proper subgraph of SG ( m, n ) has a chromatic number smaller than that of SG ( m, n ). The fractional chromatic number , χ f ( G ), of a graph G is defined as χ f ( G ) def = inf { mn | Hom(
G, KG ( m, n )) = ∅} . For more about the fractional coloring see [26]. The local chromatic number of agraph was defined in [6] as the minimum number of colors that must appear withindistance 1 of a vertex. Here is the formal definition.
Definition 1.
The local chromatic number ψ ( G ) of a graph G is ψ ( G ) def = min c max v ∈ V ( G ) |{ c ( v ) : u ∈ N ( v ) }| + 1 , where the minimum is taken over all proper colorings c of G and N ( v ) = N G ( v )denotes the neighborhood of a vertex v in a graph G . ♠ It is easy to verify that for any graph G , ψ ( G ) ≤ χ ( G ). Also, it was shown in[17] that χ f ( G ) ≤ ψ ( G ) holds for any graph G .For a graph G , let G ( k ) be the k th power of G , which is obtained on the vertexset V ( G ), by connecting any two vertices u and v for which there exists a walk oflength k between u and v in G . Note that the k th power of a simple graph is notnecessarily a simple graph itself. For instance, the k th power may have loops on itsvertices provided that k is an even integer. For a given graph G with og ( G ) ≥
7, thechromatic number of G (5) provides an upper bound for the local chromatic numberof G . In [28], it was proved ψ ( G ) ≤ ⌊ m ⌋ + 2 whenever χ ( G (5) ) ≤ m .The following simple and useful lemma was proved and used independently in[5, 25, 30]. Lemma A.
Let G and H be two simple graphs such that Hom(
G, H ) = ∅ . Thenfor any positive integer k , Hom( G ( k ) , H ( k ) ) = ∅ . Note that Lemma A trivially holds whenever H ( k ) contains a loop, e.g., when k = 2. As immediate consequences of Lemma A, we obtain χ c ( P ) = χ ( P ) andHom( C, C ) = ∅ , where P and C are the Petersen and the Coxeter graphs, respec-tively, see [5].In what follows we are concerned with some results concerning the colorings ofgraph powers. First, The notion of helical graphs is introduced. We show that suchgraphs are hom-universal with respect to high odd-girth graphs whose (2 t + 1)stpower is bounded by a Kneser graph. Then, we consider the problem of existence ofhomomorphism to odd cycles. We prove that such homomorphism to a (2 k +1)-cycleexists if and only if the chromatic number of the (2 k + 1)st power of S ( G ) is lessthan or equal to 3, where S ( G ) is the 2-subdivision of G . We also consider Neˇsetˇril’sPentagon problem. This problem is about the existence of high girth cubic graphswhich are not homomorphic to the cycle of size five. Several problems which areclosely related to Neˇsetˇril’s problem are introduced and their relations are presented.2 Helical Graphs
For a given class C of graphs, a graph U is called hom-universal with respect to C iffor any G ∈ C , Hom( G, U ) = ∅ , in which case the class C is said to be bounded bythe graph U . The problem of the existence of a bound with some special properties,for a given class of graphs, has been a subject of study in graph homomorphism. Inthe following definition, we introduce a new family of hom-universal graphs, namelythe family H ( m, n, k ) of the helical graphs. Definition 2.
Let m, n, and k be positive integers with m ≥ n . Set H ( m, n, k ) tobe the helical graph whose vertex set contains all k -tuples ( A , . . . , A k ) such that forany 1 ≤ r ≤ k , A r ⊆ [ m ] , | A | = n, | A r | ≥ n and for any s ≤ k − t ≤ k − A s ∩ A s +1 = ∅ , A t ⊆ A t +2 . Also, two vertices ( A , . . . , A k ) and ( B , . . . , B k ) of H ( m, n, k ) are adjacent if for any 1 ≤ i, j + 1 ≤ k , A i ∩ B i = ∅ , A j ⊆ B j +1 , and B j ⊆ A j +1 . ♠ Note that H ( m, ,
1) is the complete graph K m and H ( m, n,
1) is the Knesergraph KG ( m, n ). It is easy to verify that if m > n , then the odd-girth of H ( m, n, k )is greater than or equal to 2 k + 1.For a given graph G and v ∈ V ( G ), set N i ( v ) def = { u | there is a walk of length i joining u and v } . Also, for a coloring c : V ( G ) −→ (cid:0) [ m ] n (cid:1) , define c ( N i ( v )) def = [ u ∈ N i ( v ) c ( u ) . The chromatic number of graph powers has been studied in the literature (see[1, 5, 7, 23, 28, 30]). In the theorem below, we show that the helical graphs arehom-universal graphs with respect to the family of high odd-girth graphs whose(2 k − Theorem 1.
Let G be a non-empty graph with odd-girth at least k + 1 . Then wehave Hom( G (2 k − , KG ( m, n )) = ∅ if and only if Hom(
G, H ( m, n, k )) = ∅ . Proof.
First, let c ∈ Hom( G (2 k − , KG ( m, n )). If v is an isolated vertex of G ,then consider an arbitrary vertex, say f ( v ), of H ( m, n, k ). For any non-isolatedvertex v ∈ V ( G ), define f ( v ) def = ( c ( v ) , c ( N ( v )) , c ( N ( v )) , . . . , c ( N k − ( v ))) . If i ≤ j and i ≡ j mod
2, we have N i ( v ) ⊆ N j ( v ), implying that c ( N i ( v )) ⊆ c ( N j ( v )).Also, since c is a homomorphism from G (2 k − to KG ( m, n ), for any i ≤ j ≤ k − i j mod
2, we obtain c ( N i ( v )) ∩ c ( N j ( v )) = ∅ . Hence, for any vertex v ∈ V ( G ), f ( v ) ∈ V ( H ( m, n, k )). Moreover, for any 0 ≤ i, j +1 ≤ k −
1, we have N i ( v ) ∩ N i ( u ) = ∅ , N j ( v ) ⊆ N j +1 ( u ), and N j ( u ) ⊆ N j +1 ( v ) provided that u is adjacent to v . Hence, f is a graph homomorphism from G to H ( m, n, k ).3ext, let Hom( G, H ( m, n, k )) = ∅ and f : G −→ H ( m, n, k ). Assume v ∈ V ( G )and f ( v ) = ( A , A , . . . , A k ). Define, c ( v ) def = A . Assume further that u, v ∈ V ( G )such that there is a walk of length 2 t + 1 ( t ≤ k −
1) between u and v in G , i.e., uv ∈ E ( G (2 k − ). Consider adjacent vertices u ′ and v ′ such that u ′ ∈ N t ( u ) and v ′ ∈ N t ( v ).Also, let f ( v ) = ( A , A , . . . , A k ), f ( u ) = ( B , B , . . . , B k ), f ( v ′ ) = ( A ′ , A ′ , . . . , A ′ k ),and f ( u ′ ) = ( B ′ , B ′ , . . . , B ′ k ). In view of the definition of the helical graph, weobtain A ⊆ A ′ t +1 and B ⊆ B ′ t +1 . On the other hand, A ′ t +1 ∩ B ′ t +1 = ∅ , whichyields c ( v ) ∩ c ( u ) = ∅ . Thus, Hom( G (2 k − , KG ( m, n )) = ∅ , as desired. (cid:4) It was conjectured in [21] that a class C of graphs is bounded by a graph H whoseodd-girth is at least 2 k + 1 provided that the set { χ ( G (2 k − ) | G ∈ C} is bounded andthat all graphs in C have odd-girth at least 2 k + 1. It is worth noting that Theorem1 shows the above conjecture is true. This conjecture however was proved by Tardifrecently (personal communication, see [21]).It was proved by Schrijver [27] that SG ( m, n ) is the vertex-critical subgraph of KG ( m, n ). Motivated by the construction of the Schrijver graphs, we introduce afamily of subgraphs of the helical graphs. Definition 3.
Let m, n, and k be positive integers with m ≥ n . Define SG ( m, n, k )to be the induced subgraph of H ( m, n, k ) whose vertex set contains all k -tuples( A , . . . , A k ) ∈ V ( H ( m, n, k )) such that for any 1 ≤ r ≤ k , A r = ∪ s B s , where every B s is a 2-stable n -subset of [ m ]. ♠ One can deduce the following theorem whose proof is almost identical to that ofTheorem 1 and the proof is omitted for the sake of brevity.
Theorem 2.
Let G be a non-empty graph with odd-girth at least k + 1 . Then, Hom( G (2 k − , SG ( m, n )) = ∅ if and only if Hom(
G, SG ( m, n, k )) = ∅ . In [7], it was proved that χ ( H ( m, , m . Later in [1, 28], it was shown χ ( H ( m, , k )) = m . We would like to remark that the graph H ( m, , k ) is de-fined in a completely different way in [1, 28]. Simonyi and Tardos [28] showed that χ ( H ( m, , k )) = m by proving the existence of homomorphism from SG ( a, b ) to H ( m, , k ), where a − b + 2 = m and a is sufficiently large. Similarly, one can showthat χ ( H ( m, n, k )) = χ ( SG ( m, n, k )) = m − n + 2, where m ≥ n . Lemma B. [28]
Let u, v ⊂ [ a ] be two vertices of SG ( a, b ) . If there is a walk of length s between u and v in SG ( a, b ) , then | u \ v | ≤ s ( a − b + 2) . Theorem 3.
Let m, n, and k be positive integers with m ≥ n . The chromatic num-ber of the helical graph H ( m, n, k ) is equal to m − n +2 . Moreover, χ ( SG ( m, n, k )) = m − n + 2 . Proof.
For a given vertex vertex v = ( A , A , . . . , A k ) ∈ V ( H ( m, n, k )), define f ( v ) def = A . It is easy to check that f is a graph homomorphism from H ( m, n, k ) to KG ( m, n ). It follows that χ ( SG ( m, n, k )) ≤ χ ( H ( m, n, k )) ≤ m − n + 2. Now, weprove that m − n + 2 is a lower bound for the chromatic number of SG ( m, n, k ).4o this end, it suffices to show, first, that for a def = 2( k − m ( m − n + 2) + m and b def = ( k − m ( m − n + 2) + n , we have Hom( SG ( a, b ) (2 k − , SG ( m, n )) = ∅ . Then,Theorem 2 applies, and hence the assertion follows. Now, let [ a ] be partitioned into m sets, each of which contains 2( k − m − n + 2) + 1 consecutive elements of[ a ]. In other words, [ a ] is partitioned into m disjoint sets D , . . . , D m , where each D i contains consecutive elements and | D i | = 2( k − m − n + 2) + 1. Note that b = ( k − m ( m − n + 2) + n and m X i =1 ( | D i |− = ( k − m ( m − n + 2). Therefore,for every 2-stable subset u of [ a ] of size b , there are at least n indices i , . . . , i n suchthat u contains ( k − m − n + 2) + 1 elements of D i j , ≤ j ≤ n . Note also that D i contains a unique subset of cardinality ( k − m − n + 2)+ 1 which does not containany two consecutive elements. Use E i to denote this unique subset of D i , which isreadily seen to consist of the smallest elements of D i , the third smallest elements of D i , and so on and so forth. For any vertex u ∈ SG ( a, b ), we define a coloring c bychoosing n indices i j (1 ≤ j ≤ n ) such that E i j ⊆ u and we set c ( u ) def = { i , . . . , i n } .Since u is a 2-stable subset of [ a ], it is easy to verify that c ( u ) is a 2-stable subsetof [ m ] too. One needs to show that for any two vertices u and v for which there is awalk of length 2 r − ≤ r ≤ k , we have c ( u ) ∩ c ( v ) = ∅ . Toprove this, suppose that i ∈ c ( v ) and v = v , v , . . . , v r − = u be a walk between u and v , where 1 ≤ r ≤ k . By Lemma B, | v \ v r − | ≤ ( k − m − n + 2). Inparticular, v r − contains all but at most ( k − m − n + 2) elements of E i . As | E i | = ( k − m − n + 2) + 1, we see that v r − ∩ E i = ∅ . Thus, the set u , which isdisjoint from v r − , cannot contain all elements of E i , showing that i c ( u ). Thisproves that c ( u ) ∩ c ( v ) = ∅ . Therefore, Theorem 2 applies, finishing the proof. (cid:4) For a given graph G , if u and v are distinct vertices of G and the neighborhood of u isa subset of that of v , then the graph G is certainly not a vertex-critical graph. Notethat in the graph SG (7 , , { , } , { , , , } ) isa subset of that of the vertex ( { , } , { , , , , } ). Hence, the graph SG ( m, n, k )in general is not a vertex-critical graph. This motivates us to present the followingdefinition. Definition 4.
Let m, n, and k be positive integers with m ≥ n . Define SH ( m, n, k )to be the induced subgraph of H ( m, n, k ) whose vertex set contains all k -tuples( A , . . . , A k ) ∈ V ( H ( m, n, k )) such that for any 1 ≤ r ≤ k , A r = ∪ s B s and A r = ∪ t C t , where B s ’s and C t ’s are all 2-stable n -subsets of [ m ]. ♠ One can check that SH ( m, n, k ) has the property that for any two distinct vertices u, v ∈ V ( SH ( m, n, k ), N ( u ) * N ( v ) and N ( v ) * N ( u ). Also, it is straightforwardto see that SH ( m, n, k ) is the maximal subgraph of SG ( m, n, k ) with the aforemen-tioned property. To prove this, we modify the graph SG ( m, n, k ) by performing thefollowing WHILE -loop.
WHILE there exist two distinct vertices u = ( A , . . . , A k ) and v = ( B , . . . , B k ),where N ( u ) ⊆ N ( v ), then DO the following: remove the vertex u .We claim that in the WHILE-loop algorithm when the input is the graph SG ( m, n, k ) with m ≥ n , then the output is the graph SH ( m, n, k ). To show5his, note that in the WHILE -loop each time we search in the new graph forthe bad vertex u . So a vertex u may be good at the beginning, and become badlater. Suppose that WHILE-loop is not completed yet. In the last graph obtainedfrom the
WHILE-loop algorithm, let i be the greatest positive integer for whichthere exists at least a vertex u = ( A , . . . , A k ) ∈ V ( SG ( m, n, k )) such that A i isnot a union of 2-stable n -subsets of [ m ]. Note that as | A | = n , it is easy toverify that A is a union of 2-stable n -subsets of [ m ], and hence i ≥
2. Also, bythe assumption, for any i < j , A j is a union of 2-stable n -subsets of [ m ]. Set v def = ( A , . . . , A i − , A i ∪ B, A i +1 , . . . , A k ), where B def = { j | j ∈ A i and j does not appear in any 2 − stable n − subsets of A i } . For any j ∈ B , since A i − ⊆ A i and that A i − is a union of 2-stable n -subsets of[ m ], it is easy to show that { j − , j + 1 } ⊆ A i − ⊆ A i (mod m). Therefore, A i ∪ B isa union of 2-stable n -subsets of [ m ]. Also, by considering the assumption, we shouldhave B ⊆ A i +2 . Thus, v ∈ V ( SG ( m, n, k )) and also N ( u ) ⊆ N ( v ). Consequently,when the WHILE -loop is completed, we obtain the graph SH ( m, n, k ). Also, thisshows that SH ( m, n, k ) and SG ( m, n, k ) are homomorphically equivalent. In viewof the above observation, we suggest the following question. Question 1.
Let m, n, and k be positive integers with m ≥ n . Is it true that thegraph SH ( m, n, k ) is a vertex-critical graph? The problem whether the circular chromatic number and the chromatic numberof the Kneser graphs and the Schrijver graphs are equal has received attention andhas been studied in several papers [4, 9, 15, 19, 22, 28]. Johnson, Holroyd, and Stahl[15] proved that χ c (KG( m, n )) = χ (KG( m, n )) if m ≤ n + 2 or n = 2. They alsoconjectured that the equality holds for all Kneser graphs. Conjecture 1. [15]
For all m ≥ n + 1 , χ c (KG( m, n )) = χ (KG( m, n )) . It was shown in [9] that if m ≥ n ( n − m, n ) is equal to its chromatic number. Later, it was proved independentlyin [22, 28] that χ (KG( m, n )) = χ c (KG( m, n )) = m − n + 2 whenever m is an evennatural number. Also in [1, 28], it was shown that χ ( H ( m, , k )) = m . Simonyi andTardos [28] used the fact that Hom( SG ( a, b ) , H ( m, , k )) = ∅ , where a − b + 2 = m ,and hence m − H ( m, , k ).For definition of the box complex and more about this concept refer to [28]. Theorem A. ([22], [28]) If coind( B ( G )) is odd for a graph G , then χ c ( G ) ≥ coind( B ( G )) + 1 . It was shown in [28] that circular chromatic number and chromatic number of H ( m, , k ) are equal. Theorem 4.
Let m, n, and k be positive integers, where m ≥ n and m is an evenpositive integer. Then, χ c ( SG ( m, n, k )) = χ c ( H ( m, n, k )) = m − n + 2 . Further-more, χ c ( SH ( m, n, k )) = m − n + 2 . roof. As proved in Theorem 3, if a − b = m − n and a = 2( k − m ( m − n +2) + m , then Hom( SG ( a, b ) , SG ( m, n, k )) = ∅ . This implies coind( B o ( SG ( a, b )) ≤ coind( B o ( SH ( m, n, k )). Also, it is well known that coind( B o ( SG ( a, b )) = a − b + 1.Thus, by Theorem A, we have χ c ( SG ( m, n, k )) = χ c ( H ( m, n, k )) = m − n +2. Also,two graphs SH ( m, n, k ) and SG ( m, n, k ) are homomorphically equivalent. Thus, χ c ( SH ( m, n, k )) = m − n + 2. (cid:4) In [22, 28], the authors made use of Theorem A to prove that χ c (( SG ( a, b )) = χ (( SG ( a, b )) provided that a is an even positive integer. In view of χ c (( SG ( a, b )) = χ (( SG ( a, b )), where a is an even integer number, one can present an alternate proofof Theorem 4. However, note that the equality coind( B o ( H ( m, n, k )) = m − n + 1provides more information about the colorings of the helical graph H ( m, n, k ) (see[28, 29]).It was conjectured in [19] and proved in [9], that for every fixed n , there is athreshold t ( n ) such that χ c ( SG ( m, n )) = χ ( SG ( m, n )) for all m ≥ t ( n ). Note that H (3 , ,
2) is the nine cycle and that χ c ( H (3 , , . Hence, the following questionarises naturally. Question 2.
Given positive integers n and k , does there exist a number t ( n, k ) suchthat the equality χ c ( SH ( m, n, k )) = χ c ( H ( m, n, k )) = χ ( H ( m, n, k )) = m − n + 2 holds for all m ≥ t ( n, k ) ? In this section, we investigate the problem of existence of homomorphisms to oddcycles. A graph H is said to be a subdivision of a graph G if H is obtained from G by subdividing some of the edges. The graph S t ( G ) is said to be the t -subdivision ofa graph G if S t ( G ) is obtained from G by replacing each edge by a path with exactly t inner vertices. Note that S ( G ) is isomorphic to G . In the following theorem, weprove that a homomorphism to (2 k + 1)-cycle exists if and only if the chromaticnumber of (2k+1)st power of S ( G ) is less than or equal to 3. Theorem 5.
Let G be a graph with odd-girth at least k +1 . Then, χ ( S ( G ) (2 k +1) ) ≤ if and only if Hom(
G, C k +1 ) = ∅ . Proof.
First, if there exists a homomorphism from G to C k +1 , then it is obviousto see that there is a homomorphism from S ( G ) to C k +3 = H (3 , , k + 1). In viewof Theorem 1, we have χ ( S ( G ) (2 k +1) ) ≤ χ ( S ( G ) (2 k +1) ) ≤
3, then Hom( S ( G ) , C k +3 ) = ∅ . Consequently,Hom( S ( G ) (3) , C (3)6 k +3 ) = ∅ . Also, it is easy to verify that G is a subgraph of S ( G ) (3) and that there is a homomorphism from C (3)6 k +3 to C k +1 . Therefore, wehave Hom( G, C k +1 ) = ∅ . (cid:4) Considering Theorem 5, it is worth to study the following question.
Question 3.
Let G be a non-bipartite graph. What is the value of sup { k + 12 t + 1 | χ ( S t ( G ) (2 k +1) ) = χ ( G ) , k + 12 t + 1 < og ( G ) } ?7n [24], Neˇsetˇril posed the Pentagon problem. Problem 1.
Neˇsetˇril’s Pentagon Problem [24] If G is a cubic graph of sufficiently large girth, then Hom(
G, C ) = ∅ . It should be noted that if in the problem C is replaced by C , then the prob-lem holds; and in fact it is a quick consequence of Brook’s theorem. On the otherhand, the problem is known to be false if one replaces C by C , C or C [10, 18, 32].In view of Theorem 5, it is possible to rephrase the Pentagon Problem as follows. Question 4.
Let G be a cubic graph of sufficiently large girth, is it true that χ ( S ( G ) (5) ) ≤ ? If the Pentagon problem holds, then it follows from Lemma A that there existsa number g with the property that the chromatic number of the third power of anycubic graph with girth larger than g is less than six. Question 5. [5]
Is it true that for any natural number g , there exists a cubic graph G whose girth is larger than g and χ ( G (3) ) ≥ ? It is interesting to find max g ( G ) ≥ g χ ( G (3) ), where maximum is taken over all cubicgraphs with girth at least g . It should be noted that by Brook’s theorem thismaximum is less than or equal to 16. In view of Theorem 1, the following questionis equivalent to question 5. Question 6.
Is it true that for any natural number g , there exists a cubic graph G whose girth is larger than g and Hom(
G, H (5 , , ∅ ? Note that H (3 , ,
2) is the nine cycle. It was proved in [32] that the abovequestion has an affirmative answer when H (5 , ,
2) is replaced by H (3 , , Question 7.
Is it true that for any natural number g , there exists a cubic graph G whose girth is larger than g and Hom(
G, H (4 , , ∅ ? The fractional chromatic number of graphs with odd-girth greater than 3 hasbeen studied in several papers [11, 12]. Heckman and Thomas [12] posed the follow-ing conjecture.
Conjecture 2. [12]
Every triangle free graph with maximum degree at most hasthe fractional chromatic number at most . The helical graphs bound high girth graphs. Thus, it may be interesting tocompute their fractional chromatic number and their local chromatic number.
Question 8.
Let m, n, and k be positive integers with m ≥ n . What are the valuesof χ f ( H ( m, n, k )) and ψ ( H ( m, n, k ))? 8et P k +1 be the class of planar graphs of odd-girth at least 2 k + 1. Naserasr [23]posed an upper bound for the chromatic number of planar graph powers as follows. Conjecture 3. [23]
For every G ∈ P k +1 we have χ ( G (2 k − ) ≤ k . Again in view of Theorem 1, one can rephrase Naserasr’s conjecture in terms ofthe helical graphs. The following conjecture is Jaeger’s modular orientation conjec-ture restricted to planar graphs.
Conjecture 4.
Jaeger’s Conjecture [14]
Every planar graph with girth at least k has a homomorphism to C k +1 . Considering Theorem 5, one can reformulate Jaeger’s conjecture as follows.
Conjecture 5.
Let P be a planar graph with girth at least k . Then, we have χ ( S ( P ) (2 k +1) ) ≤ . Acknowledgement:
This paper was written during the sabbatical leave of theauthor in Zurich University. He wishes to thank J. Rosenthal for his hospitality.Also, the author wishes to thank an anonymous referee, G. Simonyi, C. Tardif, G.Tardos, B.R. Yahaghi, and X. Zhu who drew the author’s attention to the references[1], [7] and [12] and for their useful comments.
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