List Coloring a Cartesian Product with a Complete Bipartite Factor
aa r X i v : . [ m a t h . C O ] N ov List Coloring a Cartesian Product with a Complete BipartiteFactor
Hemanshu Kaul ∗ Jeffrey A. Mudrock † November 7, 2018
Abstract
We study the list chromatic number of the Cartesian product of any graph G anda complete bipartite graph with partite sets of size a and b , denoted χ ℓ ( G (cid:3) K a,b ). Wehave two motivations. A classic result on the gap between list chromatic number and thechromatic number tells us χ ℓ ( K a,b ) = 1 + a if and only if b ≥ a a . Since χ ℓ ( K a,b ) ≤ a for any b ∈ N , this result tells us the values of b for which χ ℓ ( K a,b ) is as large as possibleand far from χ ( K a,b ) = 2. In this paper we seek to understand when χ ℓ ( G (cid:3) K a,b ) isfar from χ ( G (cid:3) K a,b ) = max { χ ( G ) , } . It is easy to show χ ℓ ( G (cid:3) K a,b ) ≤ χ ℓ ( G ) + a . In2006, Borowiecki, Jendrol, Kr´al, and Miˇskuf showed that this bound is attainable if b is sufficiently large; specifically, χ ℓ ( G (cid:3) K a,b ) = χ ℓ ( G ) + a whenever b ≥ ( χ ℓ ( G ) + a − a | V ( G ) | . Given any graph G and a ∈ N , we wish to determine the smallest b suchthat χ ℓ ( G (cid:3) K a,b ) = χ ℓ ( G ) + a . In this paper we show that the list color function, a listanalogue of the chromatic polynomial, provides the right concept and tool for makingprogress on this problem. Using the list color function, we prove a general improvementon Borowiecki et al.’s 2006 result, and we compute the smallest such b exactly for somelarge families of chromatic-choosable graphs. Keywords. graph coloring, list coloring, Cartesian product, list color function, chromaticchoosable.
Mathematics Subject Classification.
In this paper all graphs are nonempty, finite, simple graphs unless otherwise noted. Gen-erally speaking we follow West [19] for terminology and notation. The set of natural numbersis N = { , , , . . . } . For k ∈ N , we write [ k ] for the set { , , . . . , k } . If G is a graph and S ⊆ V ( G ), we write G [ S ] for the subgraph of G induced by S . For v ∈ V ( G ), we write d G ( v )for the degree of vertex v in the graph G . If G and H are vertex disjoint graphs, we write G ∨ H for the join of G and H . ∗ Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616. E-mail: [email protected] † Department of Mathematics, College of Lake County, Grayslake, IL 60030. E-mail: [email protected] .1 List Coloring List coloring is a variation on the classic vertex coloring problem. It was introduced in the1970’s independently by Vizing [17] and Erd˝os, Rubin, and Taylor [4]. In the classic vertexcoloring problem we seek a proper k -coloring of a graph G which is a coloring of the verticesof G with colors from [ k ] such that adjacent vertices receive different colors. The chromaticnumber of a graph, denoted χ ( G ), is the smallest k such that G has a proper k -coloring.For list coloring, we associate a list assignment , L , with a graph G such that each vertex v ∈ V ( G ) is assigned a list of colors L ( v ) (we say L is a list assignment for G ). The graph G is L -colorable if there exists a proper coloring f of G such that f ( v ) ∈ L ( v ) for each v ∈ V ( G )(we refer to f as a proper L -coloring of G ). A list assignment L is called a k-assignment for G if | L ( v ) | = k for each v ∈ V ( G ). The list chromatic number of a graph G , denoted χ ℓ ( G ),is the smallest k such that G is L -colorable whenever L is a k -assignment for G . We say G is k -choosable if k ≥ χ ℓ ( G ).It is immediately obvious that for any graph G , χ ( G ) ≤ χ ℓ ( G ). Both Vizing [17] andErd˝os, Rubin, and Taylor [4] observed bipartite graphs can have arbitrarily large list chro-matic number. This implies the gap between χ ( G ) and χ ℓ ( G ) can be arbitrarily large. Thefollowing result illustrates this. Theorem 1.
For a, b ∈ N , χ ℓ ( K a,b ) = a + 1 if and only if b ≥ a a . It is worth mentioning that Theorem 1 is often attributed to Vizing [17] or Erd˝os, Rubin,and Taylor [4], but it is not stated in those papers. It is best described as a folklore result.In this paper we prove results similar in flavor to Theorem 1 for Cartesian products.Graphs in which χ ( G ) = χ ℓ ( G ) are known as chromatic-choosable graphs [12]. The notionof chromatic-choosability has received considerable attention in the literature. Many familiesof graphs have been conjectured to be chromatic-choosable (see for example [2], [6], and [7]),and there are several families of graphs that have been shown to be chromatic-choosable (seefor example [5], [8], [11], [13], and [16]). We are studying how Cartesian products with acomplete bipartite factor can be far from being chromatic-choosable. The
Cartesian product of graphs M and H , denoted M (cid:3) H , is the graph with vertex set V ( M ) × V ( H ) and edges created so that ( u, v ) is adjacent to ( u ′ , v ′ ) if and only if either u = u ′ and vv ′ ∈ E ( H ) or v = v ′ and uu ′ ∈ E ( M ). Throughout this paper, if G = M (cid:3) H and u ∈ V ( M ) (resp. u ∈ V ( H )), we let V u be the subset of V ( G ) consisting of the verticeswith first (resp. second) coordinate u . By the definition of Cartesian product of graphs, itis easy to see G [ V u ] is a copy of H (resp. M ), and we refer to G [ V u ] as the copy of H (resp. M ) corresponding to u .It is well known that χ ( G (cid:3) H ) = max { χ ( G ) , χ ( H ) } . On the other hand, the list chro-matic number of the Cartesian product of graphs is not nearly as well understood. In 2006,Borowiecki, Jendrol, Kr´al, and Miˇskuf [3] showed the following. Theorem 2 ([3]) . For any graphs G and H , χ ℓ ( G (cid:3) H ) ≤ min { χ ℓ ( G ) + col( H ) , col( G ) + χ ℓ ( H ) } − . G ), the coloring number of a graph G , is the smallest integer d for which thereexists an ordering, v , v , . . . , v n , of the elements in V ( G ) such that each vertex v i has at most d − v , v , . . . , v i − . For any graph G , it is easy to see that Theorem 2implies χ ℓ ( G (cid:3) K a,b ) ≤ χ ℓ ( G )+ a . In proving that the bound in Theorem 2 is tight, Borowiecki,Jendrol, Kr´al, and Miˇskuf also proved the following. Theorem 3 ([3]) . Suppose G is a graph with n vertices. Then, χ ℓ ( G (cid:3) K a,b ) = χ ℓ ( G ) + a whenever b ≥ ( χ ℓ ( G ) + a − an . One motivation for this paper was: For any given graph G , can we improve upon thebound on b in Theorem 3? With this question in mind, for a ∈ N , we let f a ( G ) be thesmallest b such that χ ℓ ( G (cid:3) K a,b ) = χ ℓ ( G ) + a . Along the lines of Theorem 1, computing f a ( G ) for some graph G and a ∈ N yields: χ ℓ ( G (cid:3) K a,b ) = χ ℓ ( G ) + a if and only if b ≥ f a ( G ).Theorem 1 says that f a ( K ) = a a .There are several other observations about f a ( G ) that are immediate. First, χ ℓ ( G (cid:3) K a, ) = χ ℓ ( G ) < χ ℓ ( G ) + a which implies that f a ( G ) ≥
1. Second, Theorem 3 implies that f a ( G ) ≤ ( χ ℓ ( G )+ a − a | V ( G ) | . This means f a ( G ) exists and is a natural number. Also, if G is a discon-nected graph with components: H , H , . . . , H r , we have f a ( G ) = max H i ,χ ℓ ( H i )= χ ℓ ( G ) f a ( H i ).So, we will restrict our attention to connected graphs.In this paper we use a list analogue of the chromatic polynomial called the list colorfunction to find an upper bound on f a ( G ) that is an improvement on the result of Theorem 3.We also present some further results on computing f a ( G ) in the special case where G is astrongly chromatic-choosable graph. We let P ( G, k ) be the chromatic polynomial of the graph G ; that is, P ( G, k ) is equal to thenumber of proper k -colorings of G . It can be shown that P ( G, k ) is a polynomial in k (see [1]).This notion was extended to list coloring as follows. If L is a list assignment for G , we use P ( G, L ) to denote the number of proper L -colorings of G . The list color function P ℓ ( G, k ) isthe minimum value of P ( G, L ) where the minimum is taken over all possible k -assignments L for G . Since a k -assignment could assign the same k colors to every vertex in a graph, itis clear that P ℓ ( G, k ) ≤ P ( G, k ) for each k ∈ N . In general, the list color function can differsignificantly from the chromatic polynomial for small values of k . However, Wang, Qian, andYan [18] recently showed: If G is a connected graph with m edges, then P ℓ ( G, k ) = P ( G, k )whenever k > m − √ . Also see [10] and [15] for earlier results on the list color function.In the case G is a complete graph or odd cycle, it is well known (see [14]) that P ( C n , k ) =( k − n + ( − n ( k −
1) and P ( K n , k ) = Q n − i =0 ( k − i ). It is easy to see that for each n, k ∈ N , P ( K n , k ) = P ℓ ( K n , k ), and it was recently shown in [9] that for each n, k ∈ N , P ( C n , k ) = P ℓ ( C n , k ).In [8] we introduced the notion of strong chromatic-choosability, and we used the list colorfunction to exactly compute f for graphs that are strongly chromatic-choosable. Strongchromatic-choosability is a notion of criticality in the context of chromatic-choosability. Agraph G is strong k-chromatic-choosable if χ ( G ) = k and every ( k − L , forwhich G is not L -colorable has the property that the lists are the same on all vertices . We List assignments that assign the same list of colors to every vertex of a graph are called constant . G is strongly chromatic-choosable if it is strong χ ( G )-chromatic-choosable. Note that if G is strong k -chromatic-choosable, then the only reason G is not ( k − k − G does not exist. Simple examples of strongly chromatic-choosablegraphs include complete graphs, odd cycles, and the join of a complete graph and odd cycle(see [8] for many other examples). See Section 3 for a summary of their properties, etc. Thefollowing is proven in [8]. Theorem 4 ([8]) . Let M be a strong k -chromatic-choosable graph. Then, f ( M ) = P ℓ ( M, k ) . We will generalize this result in Theorem 8, as stated in the next section.
In this subsection we present an outline of the paper while stating our results and men-tioning some open questions. Recall that f a ( G ) is defined to be the smallest b such that χ ℓ ( G (cid:3) K a,b ) = χ ℓ ( G ) + a .In Section 2, we prove that for any graph G , χ ℓ ( G (cid:3) K a,b ) = χ ℓ ( G ) + a whenever b ≥ ( P ℓ ( G, χ ℓ ( G ) + a − a . Thus, starting with any chromatic-choosable G , and taking itsCartesian product with a sequence of appropriate K a,b (with a = 0 , , , . . . ), we can constructa sequence of graphs that at each step get one farther from being chromatic-choosable: forany s ≥ t ≥ H with χ ( H ) = t and χ ℓ ( H ) = s . Theorem 5.
For any graph G and a ∈ N , f a ( G ) ≤ ( P ℓ ( G, χ ℓ ( G ) + a − a . It is easy to see that if G has at least one edge, then P ℓ ( G, χ ℓ ( G ) + a − < ( χ ℓ ( G ) + a − | V ( G ) | . This implies that Theorem 5 is an improvement on Theorem 3 whenever G has anedge. We will see many examples in Section 3 that illustrate that the bound in Theorem 5 istight (notice Theorem 4 shows the bound is tight when a = 1 and G is strongly chromatic-choosable). However, it is not the case that f a ( G ) = ( P ℓ ( G, χ ℓ ( G ) + a − a for all graphs G and a ∈ N since it is easy to see that f ( C n ) = 1, yet P ℓ ( C n ,
2) = 2. This observation leadsus to the following open question.
Question 6.
For what graphs does f a ( G ) = ( P ℓ ( G, χ ℓ ( G ) + a − a for each a ∈ N ? It is possible to slightly modify the proof idea of Theorem 5 to obtain the following moregeneral result.
Theorem 7.
Suppose H is a bipartite graph with partite sets A and B where | A | = a and | B | = b . Let δ = min v ∈ B d H ( v ) . If b ≥ ( P ℓ ( G, χ ℓ ( G ) + δ − a , then χ ℓ ( G (cid:3) H ) ≥ χ ℓ ( G ) + δ . Notice that Theorem 7 gives us conditions on when χ ℓ ( G (cid:3) H ) is guaranteed to be farfrom χ ( G (cid:3) H ) = max { χ ( G ) , } for any bipartite graph H . Furthermore, notice that when δ = a , H = K a,b . So, Theorem 7 implies Theorem 5.In Section 3 we prove that if M is a strong k -chromatic choosable graph and k ≥ a +1, then χ ℓ ( M (cid:3) K a,b ) = χ ℓ ( G ) + a if and only if b ≥ ( P ℓ ( M, χ ℓ ( M ) + a − a . This is a generalizationof Theorem 4. Theorem 8. If M is strongly chromatic-choosable and χ ( M ) ≥ a + 1 , then f a ( M ) =( P ℓ ( M, χ ℓ ( M ) + a − a . P ( K n , k ) = P ℓ ( K n , k ) and P ( C n , k ) = P ℓ ( C n , k ) whenever n, k ∈ N imply the following. Corollary 9.
The following statements hold.(i) For any l ∈ N , f ( C l +1 ) = ( P ℓ ( C l +1 , = (3 l +1 − = 9(9 l − .(ii) For n ∈ N satisfying n ≥ a + 1 , f a ( K n ) = ( P ℓ ( K n , n + a − a = (cid:16) ( n + a − a − (cid:17) a . Notice that Corollary 9 (ii) shows that the bound in Theorem 5 is tight for any a ∈ N .We do not know of any strongly chromatic-choosable graph M for which f a ( M ) < ( P ℓ ( M, χ ℓ ( M ) + a − a . This leads us to the following question.
Question 10.
Does there exist a strongly chromatic-choosable graph M such that f a ( M ) < ( P ℓ ( M, χ ℓ ( M ) + a − a ? Another interesting question involves only complete graphs.
Question 11.
Is it the case that f a ( K n ) = (cid:16) ( n + a − a − (cid:17) a for each n, a ∈ N ? Since f a ( K ) = a a , the answer to Question 11 is yes when n = 1. We have a rathertedious argument, which for the sake of brevity will not be presented in this paper, thatshows f ( K ) = 36. One could ask questions analogous to Question 11 for any family ofstrongly chromatic-choosable graphs. We end Section 3 by proving a general lower bound on f a for strongly chromatic-choosable graphs. Theorem 12.
Suppose M is a strong k -chromatic-choosable graph. Then, ( P ℓ ( M, k + a − a k − ≤ f a ( M ) . Considering Theorem 8, Theorem 12 gives us something new when χ ( M ) < a + 1. In this section we will prove Theorem 5. Before we prove this theorem, we introducesome notation and terminology that will be used for the remainder of this paper. Wheneverwe have a graph of the form H = G (cid:3) K a,b with a, b ∈ N , we will assume that the vertexset of the first factor is { v , v , . . . , v n } . We also assume that the partite sets of the copy of K a,b used to form H are { u , u , . . . , u a } and { w , w , . . . , w b } . If L is a list assignment for H = G (cid:3) K a,b and f is a proper L -coloring of H [ S aj =1 V u j ], then we say f is a bad coloringfor the copy of G corresponding to w l if there is no proper L ′ -coloring for H [ V w l ] where L ′ is the list assignment for H [ V w l ] given by L ′ ( v i , w l ) = L ( v i , w l ) − { f ( v i , u j ) : j ∈ [ a ] } for each i ∈ [ n ] . We now present a straightforward lemma related to this notion of bad coloring. We use L ( v i , w l ) rather than the technically correct L (( v i , w l )). emma 13. Suppose H = G (cid:3) K a,b with a, b ∈ N and L is a list assignment for H . Suppose C is the set of all proper L -colorings of H [ S aj =1 V u j ] . For each f ∈ C there exists an l ∈ [ b ] such that f is a bad coloring for the copy of G corresponding to w l if and only if there is noproper L -coloring of H .Proof. We prove the only if direction first. Suppose for the sake of contradiction that c is aproper L -coloring of H . Let c ′ be the proper L -coloring of H [ S aj =1 V u j ] obtained by restrictingthe domain of c to S aj =1 V u j (clearly c ′ ∈ C ). We know that there is a d ∈ [ b ] such that c ′ is abad coloring for the copy of G corresponding to w d . Let L ′ ( v i , w d ) = L ( v i , w d ) − { c ′ ( v i , u j ) : j ∈ [ a ] } for each i ∈ [ n ]. Restricting the domain of c to V w d yields a proper L ′ -coloring of thecopy of G corresponding to w d which is a contradiction.We now prove the contrapositive of the converse. Suppose there is a g ∈ C such that foreach t ∈ [ b ], g is not a bad coloring for the copy of G corresponding to w t . Let L ′ ( v i , w t ) = L ( v i , w t ) − { c ′ ( v i , u j ) : j ∈ [ a ] } for each i ∈ [ n ] and t ∈ [ b ]. For each t ∈ [ b ], since g is not abad coloring for the copy of G corresponding to w t , there is a proper L ′ -coloring of H [ V w t ].Coloring each copy of G corresponding to a vertex in { w , w , . . . , w b } according to a proper L ′ -coloring extends g to a proper L -coloring of H .We are now ready to prove Theorem 5 Proof.
Suppose H = G (cid:3) K a,b and χ ℓ ( G ) = k . Let t = P ℓ ( G, k + a − χ ℓ ( H ) = k + a when b = t a . We already know χ ℓ ( H ) ≤ k + a (byTheorem 2). So, we suppose that b = t a , and we will construct a ( k + a − L ,for H such that there is no proper L -coloring of H .Let G i be the copy of G corresponding to u i . We inductively assign lists of size ( k + a − S ai =1 V u i as follows. We begin by assigning lists, L ( v ), to each v ∈ V ( G )such that there are exactly t distinct proper L -colorings of G . Then, for each 1 < i ≤ a , weassign lists, L ( v ), to each v ∈ V ( G i ) such that there are exactly t distinct proper L -coloringsof G i and [ v ∈ V ( G i ) L ( v ) \ i − [ j =1 [ v ∈ V ( G j ) L ( v ) = ∅ (this can be done by taking the lists for G , thinking of the colors as natural numbers, andadding a sufficiently large natural number to each color in each list). Now, for i ∈ [ a ], we let c i, , c i, , . . . , c i,t denote the t distinct proper L -colorings of G i . We note that there are exactly t a proper L -colorings of H [ S ai =1 V u i ] (since for each i ∈ [ a ] we have t choices in how we color G i ). Suppose we index the t a proper L -colorings of H [ S ai =1 V u i ] as: c (1) , c (2) , . . . , c ( t a ) .Now, suppose that L ′ is a ( k − G such that there is no proper L ′ -coloringof G and [ v ∈ V ( G ) L ′ ( v ) \ a [ j =1 [ v ∈ V ( G j ) L ( v ) = ∅ . Let G ′ d be the copy of G corresponding to w d . For each d ∈ [ t a ] we assign a list, L ( v ),of size ( k + a −
1) to each v ∈ V ( G ′ d ) as follows. Suppose that the coloring c ( d ) is formedvia the colorings: c ,b , c ,b , . . . , c a,b a (note that b j is between 1 and t for each j ∈ [ a ]).6y construction, we know that |{ c j,b j ( v i , u j ) : j ∈ [ a ] }| = a for each i ∈ [ n ]. So, for each( v i , w d ) ∈ V ( G ′ d ), we let L ( v i , w d ) = L ′ ( v i ) [ { c j,b j ( v i , u j ) : j ∈ [ a ] } . This completes the construction of our ( k + a − H .Finally, notice that by construction c ( d ) is a bad coloring for G ′ d for each d ∈ [ t a ].Lemma 13 implies that there is no proper L -coloring of H .It is fairly easy to modify the idea of the proof of Theorem 5 in order to obtain a proofof Theorem 7. We would simply obtain graph H ′ from H by deleting edges in H until allvertices in B have degree δ . Then, we could use the same construction idea to obtain a( χ ℓ ( G ) + δ − G (cid:3) H ′ , L , such that there is no proper L -coloring of G (cid:3) H ′ .This would then imply χ ℓ ( G ) + δ − < χ ℓ ( G (cid:3) H ′ ) ≤ χ ℓ ( G (cid:3) H ) . f a for Strongly Chromatic-Choosable Graphs In this section we will prove Theorems 8 and 12. Suppose that M is a strong k -chromatic-choosable graph. Recall that this means χ ( M ) = k and every ( k − L , for which M is not L -colorable is constant. In this section our focus is studying f a ( M ). By Theorem 4we know that f ( M ) = P ℓ ( M, k ). So, we assume a ≥ f a ( K ) = a a for each a ∈ N ).There are several properties of strongly chromatic-choosable graphs that follow immedi-ately from the definition. We now mention some of these results (all proofs of these resultscan be found in [8]). Proposition 14 ([8]) . Suppose M is a strong k -chromatic-choosable graph. Then,(i) χ ℓ ( M ) = k (i.e. M is chromatic-choosable);(ii) If L is a list assignment for M with | L ( v ) | ≥ k − for each v ∈ V ( M ) and L is not aconstant ( k − -assignment, then there exists a proper L -coloring of M ;(iii) M ∨ K p is strong ( k + p ) -chromatic-choosable for any p ∈ N ;(iv) For any v ∈ V ( M ) , χ ( M − { v } ) ≤ χ ℓ ( M − { v } ) < k ;(v) k = 2 if and only if M is a K ;(vi) k = 3 if and only if M is an odd cycle. Proposition 15 ([8]) . Suppose M is a strong k -chromatic-choosable graph. Suppose L is anarbitrary m -assignment for M with m ≥ k . Then, for any v ∈ V ( M ) and any α ∈ L ( v ) ,there is a proper L -coloring, c , for M such that c ( v ) = α . Consequently, P ℓ ( M, m ) ≥ m max v ∈ V ( M ) P ℓ ( M − { v } , m − ≥ m. We now prove two lemmas that will lead to the proof of Theorem 8.7 emma 16.
Suppose M is a strong k -chromatic-choosable graph and H = M (cid:3) K a,b . Supposethat L is a ( k + a − -assignment for H such that there exist l, x, and y with x = y and L ( v l , u x ) ∩ L ( v l , u y ) = ∅ . Then, there is a proper L -coloring of H .Proof. Suppose that α ∈ L ( v l , u x ) ∩ L ( v l , u y ). We begin by finding a proper L -coloring foreach of the copies of M corresponding to u , . . . , u a . When it comes to the copies of M corresponding to u x and u y , we ensure that our proper L -colorings for these copies color both( v l , u x ) and ( v l , u y ) with α . We know this is possible by Proposition 15. For the remainingcopies of M , we simply take any proper L -coloring, and we know there must be at least onesuch coloring since k + a − ≥ k + 1 > k . Let c j be the proper L -coloring that we found forthe copy of M corresponding to u j for each j ∈ [ a ].Now, for i ∈ [ n ] and t ∈ [ b ] (we are defining a list assignment for the yet to be coloredvertices), we let L ′ ( v i , w t ) = L ( v i , w t ) − { c j ( v i , u j ) : j ∈ [ a ] } . It is easy to see that for each i and t , | L ′ ( v i , w t ) | ≥ k + a − − a = k −
1, and | L ′ ( v l , w t ) | ≥ k + a − − ( a −
1) = k . So, forany t ∈ [ b ], we see that L ′ restricted to the copy of M corresponding to w t is not a constant( k − L -coloring of H [ S aj =1 V u j ]is not bad for the copy of M corresponding to w t . Lemma 13 then implies there is a proper L -coloring of H . Lemma 17.
Suppose M is a strong k -chromatic-choosable graph and H = M (cid:3) K a, with k ≥ a + 1 . Suppose that L is a ( k + a − -assignment for H such that the lists L ( v i , u ) , L ( v i , u ) , . . . , L ( v i , u a ) are pairwise disjoint for each i ∈ [ n ] . Then, there is at mostone proper L -coloring of H [ S aj =1 V u j ] that is bad for the copy of M corresponding to w .Proof. Throughout this proof for each j ∈ [ a ], we let M j be the copy of M correspondingto u j . Furthermore, let M ∗ be the copy of M corresponding to w . Suppose for the sake ofcontradiction that there exist two distinct proper L -colorings, c and c ′ , of H [ S aj =1 V u i ] thatare bad for M ∗ . Since M is strong k -chromatic-choosable, the list assignments: L ′ ( v i , w ) = L ( v i , w ) − { c ( v i , u j ) : j ∈ [ a ] } and L ′′ ( v i , w ) = L ( v i , w ) − { c ′ ( v i , u j ) : j ∈ [ a ] } for i ∈ [ n ] are both constant ( k − M ∗ . Now, for j ∈ [ a ], let c j be theproper L -coloring for M j obtained by restricting the domain of c to V ( M j ), and let c a + j bethe proper L -coloring for M j obtained by restricting the domain of c ′ to V ( M j ). Since c and c ′ are different, we may assume without loss of generality that c ( v m , u ) = c a +1 ( v m , u )for some m ∈ [ n ]. Suppose c a +1 ( v m , u ) = b . Since L ( v m , u ) , L ( v m , u ) , . . . , L ( v m , u a ) arepairwise disjoint, we know that: b / ∈ a [ j =2 L ( v m , u j ) and b / ∈ { c j ( v m , u j ) : j ∈ [ a ] } . Let A be the set of ( k −
1) colors that L ′ assigns to all the vertices in M ∗ , and let B be theset of ( k −
1) colors that L ′′ assigns to all the vertices in M ∗ . We know that for i ∈ [ n ], L ( v i , w ) = A S { c j ( v i , u j ) : j ∈ [ a ] } . So, for i ∈ [ n ], B = (cid:16) A [ { c j ( v i , u j ) : j ∈ [ a ] } (cid:17) − { c a + j ( v i , u j ) : j ∈ [ a ] } . (1)8ince | B | = k − b / ∈ { c j ( v m , u j ) : j ∈ [ a ] } , and equation (1) holds for i = m , it must be thecase that b ∈ A . It immediately follows that b ∈ { c a + j ( v i , u j ) : j ∈ [ a ] } for each i ∈ [ n ]. Tosee why, note that if this did not hold we would have that some of the lists obtained from L ′′ would contain b and some would not.Now, for j ∈ [ a ], let C j = { v l : c a + j ( v l , u j ) = b } . Note C , . . . , C a are pairwise disjoint because if C r and C s both contained some vertex v p and r = s , then b ∈ L ( v p , u r ) ∩ L ( v p , u s ) which is a contradiction. Since b ∈ { c a + j ( v i , u j ) : j ∈ [ a ] } for each i ∈ [ n ], we have that V ( M ) = S aj =1 C j . Since c a + j colors all the vertices in V ( M j )with first coordinate in C j with the color b , we know that C j is an independent set of verticesin M . Thus, { C , . . . , C a } is a partition of V ( M ) into a independent sets. This implies that k = χ ( M ) ≤ a which is a contradiction. This completes the proof.We now restate and prove Theorem 8. Theorem 8. If M is strongly chromatic-choosable and χ ( M ) ≥ a + 1 , then f a ( M ) =( P ℓ ( M, χ ℓ ( M ) + a − a . Proof.
We know by Theorem 5, f a ( M ) ≤ ( P ℓ ( M, χ ℓ ( M ) + a − a . Suppose that M is strong k -chromatic-choosable with k ≥ a + 1 and H = M (cid:3) K a,b . To prove the desired, we mustshow that if b < ( P ℓ ( M, k + a − a , then χ ℓ ( H ) < k + a .Let t = P ℓ ( M, k + a − M i be the copy of M corresponding to u i , and M ′ j be the copyof M corresponding to w j . We assume that b < t a , and we let L be an arbitrary ( k + a − H . To prove the desired, we will show that there is a proper L -coloring of H .By Lemma 16, we may assume the lists L ( v i , u ) , L ( v i , u ) , . . . , L ( v i , u a ) are pairwise disjointfor each i ∈ [ n ]. For each j ∈ [ a ], there are clearly at least t distinct proper L -colorings of M j . This implies that there are at least t a proper L -colorings of H [ S aj =1 V u j ]. Let C be theset of distinct proper L -colorings of H [ S aj =1 V u j ] (we know |C| ≥ t a ).By Lemma 17 we know that for each d ∈ [ b ], there is at most one coloring in C that isbad for M ′ d . Since b < t a ≤ |C| , there must be some f ∈ C that is not a bad coloring for anyof: M ′ , . . . , M ′ b . Lemma 13 then implies there is a proper L -coloring of H .Theorem 8 along with the fact that the list color function of an odd cycle (resp. completegraph) is equal to the chromatic polynomial of the odd cycle (resp. complete graph) for eachnatural number immediately yields the following corollary. Corollary 9.
The following statements hold.(i) For any l ∈ N , f ( C l +1 ) = ( P ℓ ( C l +1 , = (3 l +1 − = 9(9 l − .(ii) For n ∈ N satisfying n ≥ a + 1 , f a ( K n ) = ( P ℓ ( K n , n + a − a = (cid:16) ( n + a − a − (cid:17) a . Notice that Corollary 9 Statement (ii) shows that the bound in Theorem 5 is tight forany a ∈ N . Suppose M is a strongly chromatic-choosable graph. At this stage of the paper,we have f a ( M ) exactly in terms of the list color function of M when χ ( M ) ≥ a + 1. When χ ( M ) < a + 1 we only have an upper bound on f a ( M ) in terms of the list color function of M (by Theorem 5). We will now turn our attention to proving Theorem 12 which will giveus a lower bound on f a ( M ) in terms of the list color function of M when χ ( M ) < a + 1. Webegin with a lemma. 9 emma 18. Suppose M is strong k -chromatic-choosable and H = M (cid:3) K a, where k < a + 1 .Let L be a ( k + a − -assignment for H such that the lists L ( v i , u ) , L ( v i , u ) , . . . , L ( v i , u a ) are pairwise disjoint for each i ∈ [ n ] . Let B be the set of proper L -colorings of H [ S aj =1 V u j ] that are bad for the copy of M corresponding to w . Then, |B| ≤ k − .Proof. We are done if B is empty. So we suppose B has at least one element. For j ∈ [ a ],let X j = { c : c ∈ L ( v , w ) ∩ L ( v , u j ) } and x j = | X j | . Note that X , X , . . . , X a must bepairwise disjoint. We claim that |B| ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a Y j =1 X j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = a Y j =1 x j where Q aj =1 X j is the Cartesian product of the sets: X , X , . . . , X a . Now, suppose that f ∈ B . Then, for i ∈ [ n ], we know that the list assignment L ′ given by L ′ ( v i , w ) = L ( v i , w ) − { f ( v i , u j ) : j ∈ [ a ] } is a constant ( k − M corresponding to w . This implies that { f ( v , u ) , f ( v , u ) , . . . , f ( v , u a ) } is a set of size a that is completely contained in L ( v , w ).Thus, ( f ( v , u ) , f ( v , u ) , . . . , f ( v , u a )) ∈ a Y j =1 X j . So, if for each f ∈ B , we let T ( f ) = ( f ( v , u ) , f ( v , u ) , . . . , f ( v , u a )), we see that T is afunction from B to Q aj =1 X j . In order to prove the desired, we will show that T is injective. Forthe sake of contradiction, suppose f and g are distinct colorings in B such that T ( f ) = T ( g ).For each i ∈ [ n ] let L ′ and L ′′ be the list assignments for the copy of M corresponding to w given by L ′ ( v i , w ) = L ( v i , w ) − { f ( v i , u j ) : j ∈ [ a ] } and L ′′ ( v i , w ) = L ( v i , w ) − { g ( v i , u j ) : j ∈ [ a ] } . We know that L ′ and L ′′ are both constant ( k −
1) assignments. Since T ( f ) = T ( g ), L ′ ( v , w ) = L ′′ ( v , w ) which immediately implies L ′ and L ′′ assign some list, A , of ( k − M corresponding to w . Since f = g there are con-stants, r and t , such that f ( v r , u t ) = g ( v r , u t ). Since L ′ and L ′′ are constant ( k − A to every vertex in the copy of M corresponding to w , we knowthat { f ( v r , u j ) : j ∈ [ a ] } = { g ( v r , u j ) : j ∈ [ a ] } . Since these two sets must have size a , f ( v r , u t ) = g ( v r , u p ) for some p = t . This however contradicts the fact that L ( v r , u t ) and L ( v r , u p ) are disjoint. Thus, T is injective, and we have |B| ≤ (cid:12)(cid:12)(cid:12)Q aj =1 X j (cid:12)(cid:12)(cid:12) .To finish the proof, we must show that Q aj =1 x j ≤ k − . Notice that each x j is a nonneg-ative integer such that P aj =1 x j ≤ k + a −
1. Under these conditions, the maximum possiblevalue of Q aj =1 x j is achieved when P aj =1 x j = k + a − x j equals ⌊ ( k + a − /a ⌋ = 1or ⌈ ( k + a − /a ⌉ = 2; that is, when ( k −
1) of x , x , . . . , x a are 2 and the rest are 1.10t is fairly easy to show that the bound in Lemma 18 is tight for each k ≥
2. For example,suppose G = K , V ( G ) = { v , v , v } , and H = G (cid:3) K , . Suppose L is the 5-assignment for H that assigns: { , , , , } to ( v i , w ) for each i ∈ [3], { , , , , } to L ( v , u ), L ( v , u ),and L ( v , u ), { , , , , } to L ( v , u ), L ( v , u ), and L ( v , u ), and { , , , , } to L ( v , u ), L ( v , u ), and L ( v , u ). It is easy to see that L satisfies the hypotheses ofLemma 18, and there are exactly 4 proper L -colorings of H [ S j =1 V u j ] that are bad for thecopy of G corresponding to w . We are now ready to restate and prove Theorem 12. Theorem 12.
Suppose M is a strong k -chromatic-choosable graph. Then, ( P ℓ ( M, k + a − a k − ≤ f a ( M ) . Proof.
Suppose that M is strong k -chromatic-choosable and H = M (cid:3) K a,b . By Theorem 8,the result is obvious when k ≥ a +1. So, we assume k < a +1. Let t = P ℓ ( M, k + a − M i be the copy of M in H corresponding to w i . We assume b < t a / k − , and we let L be anarbitrary ( k + a − H . To prove the desired, we will show there is proper L -coloring for H . By Lemma 16, we may assume that the lists L ( v i , u ) , L ( v i , u ) , . . . , L ( v i , u a )are pairwise disjoint for each i ∈ [ n ]. Let C be the set of proper L -colorings of H [ S aj =1 V u j ].Clearly, t a ≤ |C| . For d ∈ [ b ] let C d be the subset of C that contains all the proper L -colorings of H [ S aj =1 V u j ] that are bad for M d . By Lemma 18 we have that | C d | ≤ k − . Since b < t a / k − , b X d =1 | C d | ≤ b (2 k − ) < t a ≤ |C| . Thus, S bd =1 C d must be a proper subset of C , and we can find an f ∈ C − S bd =1 C d . Lemma 13then implies there is a proper L -coloring of H . References [1] G. D. Birkhoff, A determinant formula for the number of ways of coloring a map,
The Annals ofMathematics , 14 (1912), 42-46.[2] O. V. Borodin, A. V. Kostochka, and D. R. Woodall, List edge and list total colourings ofmultigraphs,
J. Combin. Theory Ser. B
71 (1997), no. 2, 184-204.[3] M. Borowiecki, S. Jendrol, D. Kr´al, and J. Miˇskuf, List coloring of cartesian products of graphs,
Discrete Mathematics
306 (2006), 1955-1958.[4] P. Erd˝os, A. L. Rubin, and H. Taylor, Choosability in graphs,
Congressus Numerantium
J. Combinatorial Theory Series B
63 (1995), no. 1, 153-158.[6] S. Gravier and F. Maffray, Choice number of 3-colorable elementary graphs,
Discrete Math.
J. Graph Theory
16 (1992), no. 5, 503-516.
8] H. Kaul and J. Mudrock, Criticality, the list color function, and list coloring the Cartesianproduct of graphs, arXiv: 1805.02147 (preprint), 2018.[9] R. Kirov and R. Naimi, List coloring and n -monophilic graphs, Ars Combinatoria
124 (2016),329-340.[10] A. V. Kostochka and A. Sidorenko, Problem session.
Fourth Czechoslovak Symposium on Com-binatorics , Prachatice, Juin (1990).[11] J. A. Noel, B. A. Reed, and H. Wu, A proof of a conjecture of ohba,
J. Graph Theory
79 (2)(2015), 86-102.[12] K. Ohba, On chromatic-choosable graphs,
J. Graph Theory
40 (2002), no. 2, 130-135.[13] A. Prowse and D.R. Woodall, Choosability of powers of circuits,
Graphs Combin.
19 (2003),137-144.[14] R. C. Read, An introduction to chromatic polynomials,
Journal of Combinatorial Theory
Journal of Combinatorial TheorySeries B
99 (2009), 474-479.[16] Zs. Tuza and M. Voigt, On a conjecture of Erd˝os, Rubin, and Taylor,
Tatra Mt. Math Publ.
Diskret. Analiz. no. 29,
MetodyDiskret. Anal. v Teorii Kodovi Skhem
Journal of Combinatorial Theory, Series B
122 (2017) 543-549.[19] D. B. West, (2001)
Introduction to Graph Theory . Upper Saddle River, NJ: Prentice Hall.. Upper Saddle River, NJ: Prentice Hall.