On the Alon-Tarsi Number and Chromatic-choosability of Cartesian Products of Graphs
aa r X i v : . [ m a t h . C O ] M a r On the Alon-Tarsi Number and Chromatic-choosability ofCartesian Products of Graphs
Hemanshu Kaul ∗ and Jeffrey A. Mudrock † Abstract
We study the list chromatic number of Cartesian products of graphs through theAlon-Tarsi number as defined by Jensen and Toft (1995) in their seminal book on graphcoloring problems. The
Alon-Tarsi number of G , AT ( G ), is the smallest k for which thereis an orientation, D , of G with max indegree k − D are different. It is known that χ ( G ) ≤ χ ℓ ( G ) ≤ χ p ( G ) ≤ AT ( G ), where χ ( G ) is the chromatic number, χ ℓ ( G ) is the list chromatic number, and χ p ( G ) is the paint number of G . In this paper we find families of graphs G and H suchthat χ ( G (cid:3) H ) = AT ( G (cid:3) H ), reducing this sequence of inequalities to equality.We show that the Alon-Tarsi number of the Cartesian product of an odd cycle and apath is always equal to 3. This result is then extended to show that if G is an odd cycle ora complete graph and H is a graph on at least two vertices containing the Hamilton path w , w , . . . , w n such that for each i , w i has a most k neighbors among w , w , . . . , w i − ,then AT ( G (cid:3) H ) ≤ ∆( G ) + k where ∆( G ) is the maximum degree of G . We discussother extensions for G (cid:3) H , where G is such that V ( G ) can be partitioned into odd cyclesand complete graphs, and H is a graph containing a Hamiltonian path. We apply thesebounds to get chromatic-choosable Cartesian products, in fact we show that these familiesof graphs have χ ( G ) = AT ( G ), improving previously known bounds. Keywords.
Cartesian product of graphs, graph coloring, list coloring, paint number,Alon-Tarsi number.
Mathematics Subject Classification.
In this paper all graphs are finite, and all graphs are either simple graphs or simpledirected graphs. Generally speaking we follow West [22] for terminology and notation. Listcoloring is a well known variation on the classic vertex coloring problem, and it was introducedindependently by Vizing [21] and Erd˝os, Rubin, and Taylor [6] in the 1970’s. In the classicvertex coloring problem we wish to color the vertices of a graph G with as few colors as possibleso that adjacent vertices receive different colors, a so-called proper coloring . The chromaticnumber of a graph, denoted χ ( G ), is the smallest k such that G has a proper coloring that ∗ Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616. E-mail: [email protected] † Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616. E-mail: [email protected] k colors. For list coloring, we associate a list assignment , L , with a graph G such thateach 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 for G ). The list chromatic number of a graph G , denoted χ ℓ ( G ), is the smallest k such that G is L -colorable whenever the listassignment L satisfies | L ( v ) | ≥ k for each v ∈ V ( G ). It is immediately obvious that for anygraph G , χ ( G ) ≤ χ ℓ ( G ). Erd˝os, Taylor, and Rubin observed in [6] that bipartite graphs canhave arbitrarily large list chromatic number. This means that the gap between χ ( G ) and χ ℓ ( G ) can be arbitrarily large, and we can not hope to find an upper bound for χ ℓ ( G ) interms of just χ ( G ).Graphs in which χ ( G ) = χ ℓ ( G ) are known as chromatic-choosable graphs (see [16]). Manyclasses of graphs have been conjectured to be chromatic-choosable. The most well knownconjecture along these lines is the List Coloring Conjecture (see [9]) which states that everyline graph of a loopless multigraph is chromatic-choosable. In addition, total graphs ([3]) andclaw free graphs ([8]) are conjectured to be chromatic-choosable. On the other hand, thereare classes of graphs that are known to be chromatic-choosable. In 1995, Galvin [7] showedthat the List Coloring Conjecture holds for line graphs of bipartite multigraphs, and in 1996,Kahn [13] proved an asymptotic version of the conjecture. Tuza and Voigt [20] showed thatchordal graphs are chromatic-choosable, and Prowse and Woodall [17] showed that powers ofcycles are chromatic-choosable. Recently, Noel, Reed, and Wu [15] proved Ohba’s conjecturewhich states that every graph, G , on at most 2 χ ( G ) + 1 vertices is chromatic-choosable.In this paper, we continue this investigation of chromatic-choosability in the realm ofCartesian Products of graphs. We study the list chromatic number of Cartesian products ofgraphs through the Alon-Tarsi number as defined by Jensen and Toft in their seminal 1995book on graph coloring problems [12]. The Alon-Tarsi number (AT-number for short) of G , AT ( G ), is the smallest k for which there is an orientation, D , of G with max indegree k − D are different. It followsfrom the Alon-Tarsi Theorem [2] that χ ( G ) ≤ χ ℓ ( G ) ≤ AT ( G ). We are interested in finding G for which these three parameters are equal.In the next two subsections, we discuss the known bounds for the list chromatic numberof the Cartesian product of graphs as well as the main tool we use to obtain our results: theAlon-Tarsi Theorem. In Section 2, we give a series of sharp bounds on the AT-number ofCartesian Products of the form G (cid:3) H , where G is such that V ( G ) can be partitioned into oddcycles and complete graphs, and H is a traceable graph (a graph containing a Hamiltonianpath). In Section 3, we apply the bounds from the previous section to give some examples ofchromatic-choosable Cartesian products of graphs, or even more strongly we show that thesegraphs have χ ( G ) = χ ℓ ( G ) = AT ( G ). The
Cartesian product of graphs G and H , denoted G (cid:3) H , is the graph with vertex set V ( G ) × 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 ( G ). Note that G (cid:3) H contains | V ( G ) | copies of H and | V ( H ) | copies of G . It is also easy to show that χ ( G (cid:3) H ) = max { χ ( G ) , χ ( H ) } . So, we havethat max { χ ( G ) , χ ( H ) } ≤ χ ℓ ( G (cid:3) H ). 2here are few results in the literature regarding the list chromatic number of the Cartesianproduct of graphs. In 2006, Borowiecki, Jendrol, Kr´al, and Miˇskuf [4] showed the following. Theorem 1 ([4]) . χ ℓ ( G (cid:3) H ) ≤ min { χ ℓ ( G ) + col( H ) , col( G ) + χ ℓ ( H ) } − . Here col( 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 atmost d − v , v , . . . , v i − . The coloring number is a classic greedy upperbound on the list chromatic number, and it immediately implies that ∆( G ) + 1 is an upperbound on the list chromatic number where ∆( G ) is the maximum degree of G . Vizing [21]extended this by proving the list coloring version of Brooks’ Theorem. Theorem 2 ([21]) . Suppose that G is a connected graph with maximum degree ∆( G ) . If G is neither a complete graph nor an odd cycle, then χ ℓ ( G ) ≤ ∆( G ) . Borowiecki et al. [4] construct examples where the upper bound in their theorem istight. Specifically, they show that if k ∈ N and G is a copy of the complete bipartitegraph K k, (2 k ) k ( k + kk ) , then χ ℓ ( G (cid:3) G ) = χ ℓ ( G ) + col( G ) −
1. On the other hand, there areexamples where the upper bounds from Theorems 1 and 2 are not tight. For example, sup-pose that G is a copy of C k +1 (cid:3) P n where n ≥
3. Since χ ℓ ( C k +1 ) = col( C k +1 ) = 3 and χ ℓ ( P n ) = col( P n ) = 2, Theorem 1 tells us that χ ℓ ( G ) ≤
4. Similarly, Theorem 2 tells us χ ℓ ( G ) ≤
4, yet we will show below that χ ℓ ( G ) = 3. Another, more dramatic, examplewhere Theorems 1 and 2 do not produce tight bounds is when we are working with theCartesian product of two complete graphs. Suppose m ≥ n ≥
2, and note that K m (cid:3) K n is the line graph for the complete bipartite graph K m,n . So, by Galvin’s celebrated result([7]): m = χ ( K m (cid:3) K n ) = χ ℓ ( K m (cid:3) K n ). However, Theorem 1 only yields an upper bound of χ ℓ ( K m (cid:3) K n ) ≤ m + n − χ ℓ ( K m ) = col( K m ) = m . Similarly, Theorem 2 only tells us χ ℓ ( K m (cid:3) K n ) ≤ m + n − G , col( G ) ≤ O ( χ ℓ ( G )) . Combining this result withTheorem 1 implies that we have an upper bound on χ ℓ ( G (cid:3) H ) in terms of only the listchromatic numbers of the factors. Borowiecki et al. [4] conjecture that a much strongerbound holds: there is a constant A such that χ ℓ ( G (cid:3) H ) ≤ A ( χ ℓ ( G ) + χ ℓ ( H )). While we willnot address this conjecture in this paper, we will present results that are improvements onTheorem 1 when the factors in the Cartesian product satisfy certain properties. Our main aimin this note is to illustrate how to utilize the classic Alon-Tarsi Theorem in these situationsto get better bounds on the AT-number and consequently the list chromatic number. Suppose graph D is a simple digraph. We say that E is a circulation contained in D if E is a spanning subgraph of D and for each v ∈ V ( D ), d − E ( v ) = d + E ( v ) (Note: d − E ( v )represents the indegree of v in E and d + E ( v ) represents the outdegree of v in E ). We saya circulation is even (resp. odd ) if it has an even (resp. odd) number of edges. Classicresults in graph theory tell us that a circulation is a digraph which consists of Euleriancomponents. This means a circulation can be decomposed into directed cycles. Alon andTarsi [2] use algebraic methods, subsequently called the Combinatorial Nullstellensatz, to3btain a remarkable relationship between a special orientation of a graph and a certain graphpolynomial. The result is the celebrated Alon-Tarsi Theorem. Theorem 3 ( Alon-Tarsi Theorem ) . Let D be an orientation of the simple graph G . Sup-pose that L is a list assignment for G such that | L ( v ) | ≥ d + D ( v ) + 1 . If the number of even andodd circulations contained in D differ, then there is a proper L -coloring for G . In addition,if the maximum indegree of D is m and the number of even and odd circulations containedin D differ, then χ ℓ ( G ) ≤ m + 1 . Recently, further implications of the Alon-Tarsi Theorem have appeared in the literature.Before mentioning one of these implications we need some terminology. For a graph G suppose that for each v ∈ V ( G ), k tokens are available at v . Two players called the marker and remover then play the following game on the graph G . For each round, the markermarks a non-empty subset, M , of vertices on the graph which uses one token for each markedvertex. The remover then selects a subset of vertices, I ⊆ M , to remove such that I is anindependent set of vertices in G . The marker wins by marking a vertex that has no tokens,and the remover wins by removing all the vertices from the graph. A graph is said to be k -paintable if the remover has a winning strategy when k tokens are available at each vertex.The paint number or online choice number of G , χ p ( G ), is the smallest k such that G is k -paintable. We have that χ ℓ ( G ) ≤ χ p ( G ), and there exist graphs where χ ℓ ( G ) < χ p ( G )(see [5] and [18]).Schauz [19] showed that if D is an orientation of G with maximum indegree m such that D satisfies the hypotheses of the Alon-Tarsi Theorem, then χ p ( G ) ≤ m + 1. So, whenever weuse the Alon-Tarsi theorem to bound the list chromatic number of a graph, we actually getthe same bound on the paint number of the graph which is a stronger result.The Alon-Tarsi number of G , AT ( G ), is the smallest k for which there is an orientation, D , of G with max indegree k − D are different (i.e. the smallest k for which the hypotheses of the Alon-Tarsi Theoremare satisfied). Jensen and Toft [12] first defined and suggested this graph invariant for study.Hefetz [11] studied the AT-number and showed, among other results, that AT ( H ) ≤ AT ( G )whenever H is a subgraph of G . Recently, Zhu [23] showed that the AT-number of all planargraphs is at most 5, improving the classic list coloring bound on planar graphs.In summary we know that: χ ( G ) ≤ χ ℓ ( G ) ≤ χ p ( G ) ≤ AT ( G ) . In general all these inequalities can be strict, our aim is to find classes of graphs where theyall are equal - a stronger form of chromatic-choosability. We study this phenomenon forCartesian products of graphs.
In this section we will start by proving that the Cartesian product of an odd cycle and pathis chromatic-choosable. Then, we prove several generalizations of this argument. Supposethat G is the Cartesian product of an arbitrary odd cycle on 2 k + 1 vertices and an arbitrarypath on n vertices. Throughout the proof of Theorem 4 we assume that we form G as4ollows. Suppose we place the vertices of C k +1 around a circle so that two vertices areadjacent if and only if they appear consecutively along the circle, and we name the verticesin counterclockwise fashion around the circle as: v , v , . . . , v k +1 (we also call this orderingof the vertices cyclic order ). Similarly we name the vertices of P n as: w , w , . . . , w n so thattwo vertices are adjacent if and only if they appear consecutively in this list (we say that thisordering of the vertices is in order ). Theorem 4.
For any k, n ∈ N , AT ( C k +1 (cid:3) P n ) = 3 . Consequently, C k +1 (cid:3) P n is chromatic-choosable.Proof. The result is obvious when n = 1. So, we assume that n ≥
2. Now, we orient the edgesof G = C k +1 (cid:3) P n as follows. For each of the n copies of C k +1 in G we orient the edges of eachcopy in counterclockwise fashion. Also we orient the edges of the form { ( v i , w j ) , ( v i , w j +1 ) } sothat ( v i , w j ) is the tail and ( v i , w j +1 ) is the head (where 1 ≤ i ≤ k +1 and 1 ≤ j ≤ n − G digraph D . We immediately note that for each ( v i , w j ) ∈ V ( D ): d − D (( v i , w j )) = ( i = 11 if i = 1 . Now, let G ∗ be the graph obtained from G by adding an extra edge in G with endpoints( v , w ) and ( v , w n ). Since G is a subgraph of G ∗ , we have that AT ( G ) ≤ AT ( G ∗ ). We willnow show that AT ( G ∗ ) ≤
3. To do this we form digraph D ∗ by orienting the edges of G ∗ sothat all the edges of G ∗ that are in G are given the same orientation as in D and the edge { ( v , w ) , ( v , w n ) } is oriented so that ( v , w ) is the head and ( v , w n ) is the tail. We callthis oriented edge e ∗ . Note that for each ( v i , w j ) ∈ V ( D ∗ ), d − D ∗ (( v i , w j )) ≤ D ∗ differ, we will provethat the number of circulations contained in D ∗ is odd. We will refer to the n oriented copiesof C k +1 in D ∗ as the base cycles . We name the base cycles: B , . . . , B n so that B i is theoriented copy of C k +1 such that all the vertices in B i have second coordinate w i . We notethat the only directed cycles contained in D ∗ are the base cycles and cycles that contain theedge e ∗ . Let C be the set of all circulations in D ∗ . We let A = (cid:26) H ∈ C (cid:12)(cid:12)(cid:12)(cid:12) H contains all the edges of at least one of the base cycles or H does not include any edge from at least one of the base cycles (cid:27) . Suppose H ∈ A . We may assume H contains all the edges from the base cycles: B a , . . . , B a m and does not include any edges from the base cycles B b , . . . , B b t (Note: at most one of thesetwo lists of base cycles may be empty). Now, we form another circulation H ′ ∈ A from H asfollows. We delete all the edges of the base cycles: B a , . . . , B a m and we add all the edges ofthe base cycles: B b , . . . , B b t . We immediately note that H = H ′ and this mapping gives usa way to pair up distinct elements of A . Thus, |A| is an even number.Now, let B = C − A . By the definition of C and A we have that B contains all thecirculations of D ∗ that contain at least one edge from each base cycle, but do not include allthe edges of any base cycle. Suppose that H ∈ B . We know that H may be decomposed intodirected cycles. No directed cycle in the decomposition of H can be a base cycle. Since theonly directed cycles in D ∗ are the base cycles and cycles containing e ∗ , we may conclude that5 is a single cycle containing e ∗ that contains at least one edge from each base cycle. Thismeans that we can view each element of B as a directed cycle starting with vertex ( v , w )and ending with the edge e ∗ .Now, for q, r ∈ Z , 0 ≤ q ≤ n −
2, and 0 ≤ r ≤ k , let e (2 k +1) q + r be the directed edge withtail ( v r +1 , w q +1 ) and head ( v r +1 , w q +2 ). For 0 ≤ i ≤ n −
2, we let: E i = { e (2 k +1) q + r | q = i and 0 ≤ r ≤ k } . Intuitively, E i consists of the directed edges from the copies of P n that connect vertices in B i +1 to vertices in B i +2 . Now, suppose that ( e a i ) n − i =0 is a subsequence of the finite sequenceof edges: ( e i ) (2 k +1)( n − − i =0 . We call ( e a i ) n − i =0 a level subsequence of edges if e a i ∈ E i for each i , a = 0, and a i a i +1 mod 2 k + 1 for each i ≤ n −
3. Now, let Q be the set of all levelsubsequences of edges, and let R be the set of all level subsequences of edges that do nothave the edge e (2 k +1)( n − as the last edge in the sequence.We will now construct a bijection between B and R . Given a sequence of edges in R ,( e a i ) n − i =0 , there is a unique cycle in D ∗ starting at ( v , w ) and ending with the edge e ∗ thatincludes each edge in ( e a i ) n − i =0 . To form this cycle, simply follow a portion of the base cycle B to get from ( v , w ) to e a . The fact that a = 0 guarantees that we must traverse atleast one edge of B . Then, for each 0 ≤ i ≤ n −
3, follow a portion of the base cycle B i +2 toget from edge e a i to e a i +1 . The fact that a i a i +1 mod 2 k + 1 for each i ≤ n − B i +2 . Finally, follow a portion of the base cycle B n to get from e a n − to e ∗ . The fact that R contains all level subsequences that do notend with the edge incident to e ∗ guarantees we must traverse at least one edge of B n . Weimmediately notice from this construction that the cycle we form contains at least one edgefrom each base cycle, but does not include all the edges of any base cycle. Thus, the cyclewe form is in B . So, we have a function from R to B . To see that this function has an inversesuppose that H ∈ B . We know that we can view H as a directed cycle starting with ( v , w )and ending with the edge e ∗ . By the way in which D ∗ is oriented we know that H mustalternate between edge(s) from B i and an edge from E i − for 1 ≤ i ≤ n −
1. Then, cycle H has edge(s) from B n followed by e ∗ . Let ( u k ) n − k =0 be the ordered sequence of edges in H that are in ∪ n − i =0 E i . We immediately have that u k ∈ E k for 0 ≤ k ≤ n − H contains at least one edge from each base cycle, u = e , and if u k = e a ′ and u k +1 = e b ′ , then a ′ b ′ mod 2 k + 1 for 0 ≤ k ≤ n −
3. So, ( u k ) n − k =0 is a levelsubsequence of edges by definition. Finally, since H contains at least one edge from B n , weknow that u n − = e (2 k +1)( n − . So, ( u k ) n − k =0 ∈ R . We now have that our function maps( u k ) n − k =0 to H and the inverse maps H to ( u k ) n − k =0 . Thus, we have a bijection between B and R and we conclude that |B| = | R | .Now, we will show that | R | is an odd number. We note that | Q | = (2 k ) n − since formingan element of Q leaves one with 2 k distinct choices at each step. We let d r,n be the numberof elements in Q that have a final edge with an index congruent to r mod 2 k + 1 (Note: n is the number of vertices in our path and 0 ≤ r ≤ k ). The following recursive relationshipsare immediate: d r,n = X i ∈{ ,..., k } ,i = r d i,n − for n ≥ d , = 0 and d i, = 1 for each i ∈ { , . . . k } . An easy inductive argument shows that6 ,n is even and d i,n is odd for each i ∈ { , . . . k } whenever n ≥
2. Since | R | = | Q | − d ,n = (2 k ) n − − d ,n , we may conclude that | R | is an odd number which immediately implies |B| is an odd number.Since |C| = |A| + |B| , we have that |C| is odd. Thus, D ∗ contains an odd number ofcirculations. This means that the number of even circulations in D ∗ does not equal thenumber of odd circulations in D ∗ . So AT ( G ∗ ) ≤ AT ( G ) ≤
3. We also have that3 ≤ χ ( G ). So, AT ( G ) = 3, and we are done. Remark.
It is easy to notice that in our proof of Theorem 4 we have proven somethingslightly stronger with regard to list coloring. Specifically we proved that if we have a listassignment, L , for G that assigns two colors to all but one vertex in the first base cycle andthree colors to all other vertices of G , then there is a proper L -coloring for G .Having proven Theorem 4, it is easy to now classify the list chromatic number of theCartesian product of an arbitrary cycle and path. Erd˝os et al. [6] classified all graphs withlist chromatic number equal to 2. Let Θ( l , . . . , l k ) with branch vertices u and v be the graphthat is the union of k pairwise internally disjoint u, v -paths of lengths l , . . . , l k . Theorem 5 ([6]) . Let G be a connected bipartite graph. Then, χ ℓ ( G ) = 2 if and only if G has at most one cycle or the subgraph consisting of the non-cut-edges of G is Θ(2 , , t ) forsome t ∈ N . This is the final ingredient in:
Corollary 6.
For k ∈ N :(i) χ ( C k +1 (cid:3) P n ) = χ ℓ ( C k +1 (cid:3) P n ) = 3 for n ∈ N ,(ii) χ ( C k +2 (cid:3) P ) = χ ℓ ( C k +2 (cid:3) P ) = 2 , and χ ℓ ( C k +2 (cid:3) P n ) = 3 for n ≥ .Proof. Statement (i) follows from Theorem 4. For statement (ii) notice that when n ≥ C k +2 (cid:3) P n contains more than one cycle and no cut edges. Moreover, all the vertices in C k +2 (cid:3) P n have degree at least 3 which means that C k +2 (cid:3) P n is not Θ(2 , , t ) for any t ∈ N .So, Theorem 5 implies that 3 ≤ χ ℓ ( C k +2 (cid:3) P n ). By Theorem 5, we also have that even cycleshave list chromatic number equal to 2. So, Theorem 1 implies χ ℓ ( C k +2 (cid:3) P n ) ≤ − . We now give a natural generalization of the argument of Theorem 4.
Theorem 7.
Suppose that G is a complete graph or an odd cycle with | V ( G ) | ≥ . Suppose H is a graph on at least two vertices that contains a Hamilton path, w , w , . . . , w m , suchthat w i has at most k neighbors among w , . . . , w i − . Then, AT ( G (cid:3) H ) ≤ ∆( G ) + k . Before we prove this theorem a couple of remarks are worth making. We require m ≥ χ ℓ ( G (cid:3) H ) = ∆( G ) + 1 when m = 1. Now, suppose G and H satisfy thehypotheses of the Theorem. We have that χ ℓ ( G ) = col( G ) = ∆( G ) + 1 and col( H ) − ≤ k .Theorem 1 tells us that χ ℓ ( G (cid:3) H ) ≤ ∆( G ) + χ ℓ ( H ), and Theorem 2 implies that χ ℓ ( G (cid:3) H ) ≤ ∆( G ) + ∆( H ). So, Theorem 7 gives us an improvement on these known bounds if and only7f k < χ ℓ ( H ) and k < ∆( H ). It is easy to see that k < χ ℓ ( H ) and k < ∆( H ) if and only if k = col( H ) − χ ℓ ( H ) − d H ( w m ) = ∆( H ). We show examples in Section 3 whereTheorem 7 improves these known bounds. We now present the proof. Proof.
Suppose G has n vertices. We name the vertices of G (cyclically if G is a cycle): v , v , . . . , v n . G contains an odd cycle, C , as an induced subgraph. If G is an odd cycle let C = G , and if G is a complete graph let C be the subgraph of G induced by the vertices v n − , v n − , and v n . For the remainder of the proof assume that C has 2 k + 1 vertices where k ∈ N (Note: we know k = 1 when G is a complete graph). We also let P be the Hamiltonpath, w , w , . . . , w m , contained in H .Now, consider the graph G (cid:3) H . We form digraph D from this graph by orienting its edgesas follows. We begin by orienting each of the m copies of C in a counterclockwise fashion.Then, for each edge in a copy of G and not in a copy of C with endpoints ( v r , w u ) and ( v s , w u )with s > r , we orient the edge so that ( v r , w u ) is the tail and ( v s , w u ) is the head. Finally,for each edge in a copy of H with endpoints ( v u , w r ) and ( v u , w s ) with s > r , we orient theedge so that ( v u , w r ) is the tail and ( v u , w s ) is the head.From D we form the digraph D ∗ by adding a directed edge with tail ( v n − k +1 , w m ) andhead ( v n − k , w ). Note that D ∗ is a simple digraph since m ≥
2. We will refer to the edge weadded as e ∗ . We immediately note that by the conditions placed on G and H , we have that d − D ∗ (( v r , w u )) ≤ ∆( G ) − k for each ( v r , w u ) ∈ V ( D ∗ ). Similar to the proof of Theorem 4,we will now prove that the number of circulations in D ∗ is odd. We will refer to the m oriented copies of C in D ∗ as the base cycles . We name the base cycles: B , . . . , B m so that B i is the oriented copy of C such that all the vertices in B i have second coordinate w i . Wenote that the only cycles in D ∗ are the base cycles and the cycles that contain e ∗ . Let S be the subgraph of D ∗ that is made up of the m oriented copies of C in D ∗ , the orientedcopies of P that have first coordinate v n − k , v n − k +1 , . . . , and v n , and the edge e ∗ . Noticethat S is C (cid:3) P plus edge e ∗ oriented as it is in the proof of Theorem 4. Let C be the set ofall circulations in D ∗ . We define A as in the proof of Theorem 4. By the same argument asin the proof of Theorem 4, we see that |A| is even. Now, let B = C − A . As in the proof ofTheorem 4, we have that any K ∈ B is a single cycle containing e ∗ that contains at least oneedge from each base cycle. Moreover, any K ∈ B must be completely contained in S (sinceany vertex of the form ( v ℓ , w u ) with l < n − k is in no cycles of D ∗ and any cycle in D ∗ thatcontains an oriented edge of a copy of H that is not in a copy of P must leave out at leastone base cycle).Thus, by the same argument as in the proof of Theorem 4, we have that |B| is odd. Thus, |C| is odd, and we have the desired result by the Alon-Tarsi Theorem. Remark.
Note that our proof also works when G is any graph that contains an inducedodd cycle C such that every maximum degree vertex in G is either in C or adjacent to C .However, this is not particularly useful for bounding χ ℓ ( G (cid:3) H ), since when G is not an oddcycle or complete graph Theorems 2 and 1 yield: χ ℓ ( G (cid:3) H ) ≤ χ ℓ ( G ) + col( H ) − ≤ ∆( G ) + k + 1 − G ) + k. Now, we present another extension of Theorem 4. We first need some definitions andnotation. Suppose that G is an arbitrary graph, and T is some subset of V ( G ). We write8 [ T ] for the subgraph of G induced by the vertices in T . Now, suppose that G and G aretwo arbitrary vertex disjoint graphs. The join of the graphs G and G , denoted G ∨ G ,is the graph consisting of G , G , and additional edges added so that each vertex in G is adjacent to each vertex in G . When a graph, G , consists of G , G , and some set ofadditional edges (possibly empty) that have one endpoint in V ( G ) and the other endpointin V ( G ) we say that G is a partial join of G and G . Theorem 8.
Suppose that G is a graph with the property that there exists a partition, { S , . . . , S m } , of V ( G ) such that for each i G [ S i ] is an odd cycle. For each i ≥ sup-pose each vertex in S i has at most ρ neighbors in ∪ i − j =1 S j (we let ρ = 0 in the case that m = 1 ). Then, AT ( G (cid:3) P k ) ≤ ρ , for any k ∈ N . Note that we obtain Theorem 4 when m = 1. We also notice that when m = 2, G is apartial join of two odd cycles. When m = 3, G is a partial join of the odd cycle G [ S ] and G [ S ∪ S ] (where G [ S ∪ S ] is itself the partial join of two odd cycles). And so on. Proof.
First, for each i , we index the vertices of G [ S i ] in cyclic order as: v i, , v i, , . . . , v i,m i where we know that m i is odd for each i . If e ∈ E ( G ) is an edge with one endpoint in S i andone endpoint in S j with i = j we refer to e as a connecting edge .We now turn our attention to the case where k = 1. In this case, we must show that AT ( G ) ≤ ρ . We form an oriented version, D , of the graph G as follows. For each oddcycle, G [ S i ], orient the edges of the cycle so that v i,l is the tail and v i,l +1 is the head for1 ≤ l ≤ m i −
1. Then, orient the final edge of the cycle so that v i, is the tail and v i,m i is thehead. Finally, for each connecting edge, e ∈ E ( G ), with one endpoint in S i and one endpointin S j with i < j orient e so that its tail is in S i and its head is in S j . One may now note thatfor each v ∈ S i , d − D ( v ) ≤ ρ. We now claim that D is acyclic. First, note that no oriented connecting edge can be in acycle in D since there is no way in D to get from a vertex in S j to a vertex in S i when i < j .This means that if there is a cycle, C , in D there must exist an i such that the vertices of C are a subset of the vertices of S i . However, D [ S i ] is acyclic by construction. So, no such C can exist. This means that D has one even circulation (the circulation with no edges) andzero odd circulations. Then AT ( G ) ≤ ρ as desired.We now turn our attention to the case where k ≥
2. Let H = P k . We note that G (cid:3) H ismade of the disjoint union of graphs: m X i =1 ( G [ S i ] (cid:3) H )plus the connecting edges in each of the k copies of G . Let G ∗∗ be the graph formed from G (cid:3) H by adding an edge in G [ S i ] (cid:3) H for each i as we did in the proof of Theorem 4 to form G ∗ . We call the newly formed subgraph of G ∗∗ consisting of G [ S i ] (cid:3) H plus an additionaledge, M i for each i . We form an oriented version, D , of G ∗∗ as follows. For each subgraph M i of G ∗∗ , we orient this subgraph just as we oriented D ∗ in the proof of Theorem 4. Finally,we orient all the connecting edges in all the copies of G just as we did in the case of k = 1.9ne may note that for each v ∈ V ( D ), we have that d − D ( v ) ≤ ρ . Similar to the case where k = 1, we note that no connecting edge from any copy of G is in a cycle contained in D . Thismeans that if C i is the number of circulations in the oriented version of M i , the number ofcirculations in D is equal to m Y i =1 C i which by the proof of Theorem 4 is an odd number. Thus, the number of even circulationsin D does not equal the number of odd circulations in D , and we have that AT ( G (cid:3) H ) ≤ AT ( G ∗∗ ) ≤ ρ + 3 . Finally, we can easily combine the idea of the proofs of Theorems 7 and 8 to obtain thefollowing.
Theorem 9.
Suppose that G is a graph with the property that there exists a partition, { S , . . . , S m } , of V ( G ) such that for each i | S i | ≥ and G [ S i ] is an odd cycle or a com-plete graph. For each i ≥ suppose each vertex in S i has at most ρ i neighbors in ∪ i − j =1 S j ,and let ρ = 0 . For each i ≥ we let: α i = ( ρ i + 3 if G [ S i ] is an odd cycle ρ i + | S i | if G [ S i ] is a complete graph.Now, let α = max i α i . Suppose H is a graph on at least two vertices that contains a Hamil-ton path, w , w , . . . , w n , such that w i has at most k neighbors among w , . . . , w i − . Then, AT ( G ) ≤ α and AT ( G (cid:3) H ) ≤ α + k − . Both Theorems 8 and 9 are sharp and give improvements over existing bounds as shownin examples in the next section.
In this section we present some examples where our results from Section 2 improve uponknown bounds for the list chromatic number. We let the kth power of graph G , denoted G k ,be the graph with vertex set V ( G ) where two vertices are adjacent if their distance in G is atmost k . It is easy to see that P rn , 1 ≤ r ≤ n −
1, satisfies χ ( P rn ) = χ ℓ ( P rn ) = col( P rn ) = r + 1.We already know Theorem 7 is sharp by r = 1 below. Corollary 10.
Suppose that k, r, n ∈ N are such that n ≥ and r ≤ n − .Then, max { r + 1 , } ≤ AT ( C k +1 (cid:3) P rn ) ≤ r + 2 .Proof. Let G = C k +1 and H = P rn . Name the vertices of H : w , w , . . . , w n so that w i is the i th vertex of the underlying path on n vertices contained in H . Clearly, H contains the Hamil-ton path: w , w , . . . , w n , and for each i , w i has at most r neighbors among w , w , . . . , w i − .Thus, by Theorem 7, we have that AT ( G (cid:3) H ) ≤ r .We note that Theorem 1 only yields that χ ℓ ( G (cid:3) H ) ≤ r + 3. Also, Theorem 2 yields χ ℓ ( G (cid:3) H ) ≤ P rn ), and r + 1 ≤ ∆( P rn ) when n ≥ r ≤ n −
2. So, the above example10mproves upon known bounds on the list chromatic number when n ≥ r ≤ n −
2. Wesuspect that C k +1 (cid:3) P rn is often chromatic-choosable, but improving upon our upper boundwith the Alon-Tarsi Theorem seems difficult. For example, it would be impossible to find anorientation of C k +1 (cid:3) P n with max indegree of 2 for large values of n .Before we move on to an example for Theorems 8 and 9, let us note a couple of other factsimplied by Theorem 7: AT ( K n (cid:3) P m ) = n , for n ≥
3; and n − ≤ AT ( C k +1 (cid:3) H n ) ≤ n − H n = K n − E ( P ) for n ≥ Corollary 11.
For G = K m ∨ C k +1 , AT ( G (cid:3) P n ) = m + 3 , where m, k, n ∈ N . Consequently, ( K m ∨ C k +1 ) (cid:3) P n is chromatic-choosable.Proof. The result is obvious when n = 1 since col( K m ∨ C k +1 ) = m + 3. So, suppose n ≥
2. Let G = K m and G = C k +1 . Since χ ( G ∨ G ) = χ ( G ) + χ ( G ), we have that m + 3 = χ ( G ) = χ ( G (cid:3) P n ).First, consider the case where m ≥
3. Let { S , S } be the partition of V ( G ) where S = V ( G ) and S = V ( G ). Note G [ S ] is a complete graph, and G [ S ] is an odd cycle.Also, each vertex in S has exactly m neighbors in S . So, by Theorem 9 we have that AT ( G (cid:3) P n ) ≤ m + 3.Finally, suppose m = 1 ,
2. Let G ( m ) be a partial join of G = C and G so that m vertices in V ( G ) are adjacent to all the vertices in V ( G ), and the other 3 − m vertices in V ( G ) are not adjacent to any of the vertices in V ( G ). Now, let { S , S } be the partitionof V ( G ( m ) ) where S = V ( G ) and S = V ( G ). Note G ( m ) [ S ] and G ( m ) [ S ] are odd cycles,and each vertex in S has exactly m neighbors in S . So, by Theorem 8 we have that AT ( G ( m ) (cid:3) P n ) ≤ m + 3. The result follows since G (cid:3) P n is a subgraph of G ( m ) (cid:3) P n .We note that when n ≥ χ ℓ (( K m ∨ C k +1 ) (cid:3) P n ) ≤ m + 4.Also, when n ≥
3, Theorem 2 only tells us χ ℓ (( K m ∨ C k +1 ) (cid:3) P n ) ≤ max { m + 4 , m + 2 k + 2 } .There is still much to be discovered about the list chromatic number of the Cartesianproduct of graphs. Aside from the important conjectures proposed in [4], this paper alsoprovides us with some interesting questions. Specifically, an ambitious question would be:Can we determine when G (cid:3) H will be chromatic-choosable based upon some property ofthe factors? We further explore this question in a following paper [14]. Simpler questions,motivated by Corollaries 6 and 10, include: For what graphs G , is G (cid:3) P n chromatic-choosable?When is C k +1 (cid:3) P rn chromatic-choosable? For the first of these questions one may conjecturebased upon the results of this paper that if G is chromatic-choosable and χ ( G ) ≥
3, then G (cid:3) P n chromatic-choosable. However, this conjecture is false since one can construct a 3-assignment to show that 3 < χ ℓ ( C (cid:3) P ). References [1] N. Alon, Degrees and choice numbers,
Random Structures Algorithms
16 (2000), 364-368.[2] N. Alon and M. Tarsi, Colorings and orientations of graphs,
Combinatorica
12 (1992), 125-134.[3] 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.
4] M. Borowiecki, S. Jendrol, D. Kr´al, and Miˇskuf, List coloring of cartesian products of graphs,
Discrete Mathematics
306 (2006), 1955-1958.[5] J. Carraher, S. Loeb, T. Mahoney, G. Puleo, M. Tsai, and D. B. West, Three topics in onlinelist coloring,
Journal of Combinatorics
Congressus Numerantium
J. Combinatorial Theory Series B
63 (1995), no. 1, 153-158.[8] S. Gravier, and F. Maffray, Choice number of 3-colorable elementary graphs,
Discrete Math.
J. Graph Theory
16 (1992), no. 5, 503-516.[10] Haj´os, G., ¨Uber eine Konstruktion nicht n -f¨arbbarer graphen, Wiss. Z. Martin-Luther-Univ.Halle-Wittenberg Math.-Natur. Reihe
10 (1961), 116-117.[11] D. Hefetz, On two generalizations of the Alon-Tarsi polynomial method,
J. Combin. Theory Ser.B
101 (2011), no. 6, 403-414.[12] T. Jensen, B. Toft, Graph Coloring Problems, Wiley, New York, 1995.[13] J. Kahn, Asymptotically good list-colorings,
J. Combin. Theory Ser. A
73 (1996), no. 1, 1-59.[14] H. Kaul, J. Mudrock, Criticality, List Color Function, and List Coloring the Cartesian Productof Graphs,
Preprint , 2018.[15] J. A. Noel, B. A. Reed, H. Wu, A proof of a conjecture of Ohba,
J. Graph Theory
79 (2) (2015),86-102.[16] K. Ohba, On chromatic-choosable graphs,
J. Graph Theory
40 (2002), no. 2, 130-135.[17] A. Prowse and D.R. Woodall, Choosability of powers of circuits,
Graphs Combin.
19 (2003),137-144.[18] U. Schauz, Mr. Paint and Mrs. Correct,
The Electronic Journal of Combinatorics
16 (2009), no.1, R77.[19] U. Schauz, Flexible lists in Alon and Tarsi’s theorem, and time scheduling with unreliable par-ticipants,
The Electronic Journal of Combinatorics
17 (2010), no. 1, R13.[20] 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
Introduction to Graph Theory . Upper Saddle River, NJ: Prentice Hall.[23] X. Zhu, The Alon-Tarsi number of planar graphs, arXiv:1711.10817 submitted 2017.submitted 2017.