Circular choosability

We study circular choosability, a notion recently introduced by Mohar and Zhu. First, we provide a negative answer to a question of Zhu about circular cliques. We next prove that cch(G)=O(ch(G)+ln|V(G)|) for every graph G. We investigate a generalization of circular choosability, the circular f‐choosability, where f is a function of the degrees. We also consider the circular choice number of planar graphs. Mohar asked for the value of τ ? sup{cch(G) : G is planar}, and we prove that 6≤τ≤8, thereby providing a negative answer to another question of Mohar. We also study the circular choice number of planar and outerplanar graphs with prescribed girth, and graphs with bounded density. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 241–270, 2009     

Circular consecutive choosability of k‐choosable graphs

Let S(r) denote a circle of circumference r. The circular consecutive choosability ch_cc(G) of a graph G is the least real number t such that for any r≥χ_c(G), if each vertex v is assigned a closed interval L(v) of length t on S(r), then there is a circular r‐coloring f of G such that f(v)∈L(v). We investigate, for a graph, the relations between its circular consecutive choosability and choosability. It is proved that for any positive integer k, if a graph G is k‐choosable, then ch_cc(G)?k + 1 ? 1/k; moreover, the bound is sharp for k≥3. For k = 2, it is proved that if G is 2‐choosable then ch_cc(G)?2, while the equality holds if and only if G contains a cycle. In addition, we prove that there exist circular consecutive 2‐choosable graphs which are not 2‐choosable. In particular, it is shown that ch_cc(G) = 2 holds for all cycles and for K_{2, n} with n≥2. On the other hand, we prove that ch_cc(G)>2 holds for many generalized theta graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory 67: 178‐197, 2011     

Circular coloring of signed graphs

Let ( be two positive integers. We generalize the well‐studied notions of ‐colorings and of the circular chromatic number to signed graphs. This implies a new notion of colorings of signed graphs, and the corresponding chromatic number χ. Some basic facts on circular colorings of signed graphs and on the circular chromatic number are proved, and differences to the results on unsigned graphs are analyzed. In particular, we show that the difference between the circular chromatic number and the chromatic number of a signed graph is at most 1. Indeed, there are signed graphs where the difference is 1. On the other hand, for a signed graph on n vertices, if the difference is smaller than 1, then there exists , such that the difference is at most . We also show that the notion of ‐colorings is equivalent to r‐colorings (see [12] (X. Zhu, Recent developments in circular coloring of graphs, in Topics in Discrete Mathematics Algorithms and Combinatorics Volume 26 , Springer Berlin Heidelberg, 2006, pp. 497–550)).     

Circular perfect graphs

For 1 ≤ d ≤ k, let K_k/d be the graph with vertices 0, 1, …, k ? 1, in which i ～j if d ≤ |i ? j| ≤ k ? d. The circular chromatic number χ_c(G) of a graph G is the minimum of those k/d for which G admits a homomorphism to K_k/d. The circular clique number ω_c(G) of G is the maximum of those k/d for which K_k/d admits a homomorphism to G. A graph G is circular perfect if for every induced subgraph H of G, we have χ_c(H) = ω_c(H). In this paper, we prove that if G is circular perfect then for every vertex x of G, N_G[x] is a perfect graph. Conversely, we prove that if for every vertex x of G, N_G[x] is a perfect graph and G ? N[x] is a bipartite graph with no induced P₅ (the path with five vertices), then G is a circular perfect graph. In a companion paper, we apply the main result of this paper to prove an analog of Haj?os theorem for circular chromatic number for k/d ≥ 3. Namely, we shall design a few graph operations and prove that for any k/d ≥ 3, starting from the graph K_k/d, one can construct all graphs of circular chromatic number at least k/d by repeatedly applying these graph operations. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 186–209, 2005     

Circular choosability via combinatorial Nullstellensatz

A p‐list assignment L of a graph G assigns to each vertex v of G a set of permissible colors. We say G is L‐(P, q)‐colorable if G has a (P, q)‐coloring h such that h(v) ? L(v) for each vertex v. The circular list chromatic number of a graph G is the infimum of those real numbers t for which the following holds: For any P, q, for any P‐list assignment L with , G is L‐(P, q)‐colorable. We prove that if G has an orientation D which has no odd directed cycles, and L is a P‐list assignment of G such that for each vertex v, , then G is L‐(P, q)‐colorable. This implies that if G is a bipartite graph, then , where is the maximum average degree of a subgraph of G. We further prove that if G is a connected bipartite graph which is not a tree, then . © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 190–204, 2008     

Circular chromatic index of Cartesian products of graphs

The circular chromatic index of a graph G, written , is the minimum r permitting a function such that whenever e and are incident. Let □ , where □ denotes Cartesian product and H is an ‐regular graph of odd order, with (thus, G is s‐regular). We prove that , where is the minimum, over all bases of the cycle space of H, of the maximum length of a cycle in the basis. When and m is large, the lower bound is sharp. In particular, if , then □ , independent of m. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 7–18, 2008     

Total choosability of multicircuits II

A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2‐regular graph). In this paper, the method of Alon and Tarsi is used to prove that all multicircuits of even order, and some regular and near‐regular multicircuits of odd order have total choosability (i.e., list total chromatic number) equal to their ordinary total chromatic number. This completes the proof that every multicircuit has total choosability equal to its total chromatic number. In the process, the total chromatic numbers of all multicircuits are determined. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 44–67, 2002     

Structural properties and edge choosability of planar graphs without 4-cycles

Some structural properties of planar graphs without 4-cycles are investigated. By the structural properties, it is proved that every planar graph G without 4-cycles is edge-(Δ(G)+1)-choosable, which perfects the result given by Zhang and Wu: If G is a planar graph without 4-cycles, then G is edge-t-choosable, where t=7 if Δ(G)=5, and otherwise t=Δ(G)+1.     

Total choosability of multicircuits I

A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2‐regular graph). In this pair of papers, it is proved that every multicircuit C has total choosability (i.e., list total chromatic number) ch′′(C) equal to its ordinary total chromatic number χ′′(C). In the present paper, the kernel method is used to prove this for every multicircuit that has at least two vertices with degree less than its maximum degree Δ. The result is also proved for every multicircuit C for which χ′′(C)≥Δ+2. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 26–43, 2002     

Game chromatic index of k‐degenerate graphs

We consider the following edge coloring game on a graph G. Given t distinct colors, two players Alice and Bob, with Alice moving first, alternately select an uncolored edge e of G and assign it a color different from the colors of edges adjacent to e. Bob wins if, at any stage of the game, there is an uncolored edge adjacent to colored edges in all t colors; otherwise Alice wins. Note that when Alice wins, all edges of G are properly colored. The game chromatic index of a graph G is the minimum number of colors for which Alice has a winning strategy. In this paper, we study the edge coloring game on k‐degenerate graphs. We prove that the game chromatic index of a k‐degenerate graph is at most Δ + 3k − 1, where Δ is the maximum vertex degree of the graph. We also show that the game chromatic index of a forest of maximum degree 3 is at most 4 when the forest contains an odd number of edges. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 144–155, 2001     

Circular game chromatic number of graphs

In a circular r-colouring game on G, Alice and Bob take turns colouring the vertices of G with colours from the circle S(r) of perimeter r. Colours assigned to adjacent vertices need to have distance at least 1 in S(r). Alice wins the game if all vertices are coloured, and Bob wins the game if some uncoloured vertices have no legal colour. The circular game chromatic number χ_cg(G) of G is the infimum of those real numbers r for which Alice has a winning strategy in the circular r-colouring game on G. This paper proves that for any graph G, , where is the game colouring number of G. This upper bound is shown to be sharp for forests. It is also shown that for any graph G, χ_cg(G)≤2χ_a(G)(χ_a(G)+1), where χ_a(G) is the acyclic chromatic number of G. We also determine the exact value of the circular game chromatic number of some special graphs, including complete graphs, paths, and cycles.     

The circular chromatic number of series‐parallel graphs

In this article, we consider the circular chromatic number χ_c(G) of series‐parallel graphs G. It is well known that series‐parallel graphs have chromatic number at most 3. Hence, their circular chromatic numbers are at most 3. If a series‐parallel graph G contains a triangle, then both the chromatic number and the circular chromatic number of G are indeed equal to 3. We shall show that if a series‐parallel graph G has girth at least 2 ⌊(3k − 1)/2⌋, then χ_c(G) ≤ 4k/(2k − 1). The special case k = 2 of this result implies that a triangle free series‐parallel graph G has circular chromatic number at most 8/3. Therefore, the circular chromatic number of a series‐parallel graph (and of a K₄‐minor free graph) is either 3 or at most 8/3. This is in sharp contrast to recent results of Moser [5] and Zhu [14], which imply that the circular chromatic number of K₅‐minor free graphs are precisely all rational numbers in the interval [2, 4]. We shall also construct examples to demonstrate the sharpness of the bound given in this article. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 14–24, 2000     

Circular chromatic number and Mycielski construction

This paper gives a sufficient condition for a graph G to have its circular chromatic number equal to its chromatic number. By using this result, we prove that for any integer t ≥ 1, there exists an integer n such that for all . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 106–115, 2003     

The circular chromatic number of series‐parallel graphs with large girth

It was proved by Hell and Zhu that, if G is a series‐parallel graph of girth at least 2⌊(3k − 1)/2⌋, then χ_c(G) ≤ 4k/(2k − 1). In this article, we prove that the girth requirement is sharp, i.e., for any k ≥ 2, there is a series‐parallel graph G of girth 2⌊(3k − 1)/2⌋ − 1 such that χ_c(G) > 4k/(2k − 1). © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 185–198, 2000     

Total choosability of planar graphs with maximum degree 4

Let G be a planar graph with maximum degree 4. It is known that G is 8-totally choosable. It has been recently proved that if G has girth g?6, then G is 5-totally choosable. In this note we improve the first result by showing that G is 7-totally choosable and complete the latter one by showing that G is 6-totally choosable if G has girth at least 5.     

Circular choosability is rational

The circular choosability or circular list chromatic number of a graph is a list-version of the circular chromatic number, that was introduced by Mohar in 2002 and has been studied by several groups of authors since then. One of the nice properties that the circular chromatic number enjoys is that it is a rational number for all finite graphs G, and a fundamental question, posed by Zhu and reiterated by others, is whether the same holds for the circular choosability. In this paper we show that this is indeed the case.     

On two questions about circular choosability

We answer two questions of Zhu on circular choosability of graphs. We show that the circular list chromatic number of an even cycle is equal to 2 and give an example of a graph for which the infimum in the definition of the circular list chromatic number is not attained. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:261‐269, 2008     

Weight choosability of graphs

Suppose the edges of a graph G are assigned 3‐element lists of real weights. Is it possible to choose a weight for each edge from its list so that the sums of weights around adjacent vertices were different? We prove that the answer is positive for several classes of graphs, including complete graphs, complete bipartite graphs, and trees (except K₂). The argument is algebraic and uses permanents of matrices and Combinatorial Nullstellensatz. We also consider a directed version of the problem. We prove by an elementary argument that for digraphs the answer to the above question is positive even with lists of size two. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 242–256, 2009