Edge‐face chromatic number and edge chromatic number of simple plane graphs

Given a simple plane graph G, an edge‐face k‐coloring of G is a function ? : E(G) ∪ F(G) → {1,…,k} such that, for any two adjacent or incident elements a, b ∈ E(G) ∪ F(G), ?(a) ≠ ?(b). Let χ_e(G), χ_ef(G), and Δ(G) denote the edge chromatic number, the edge‐face chromatic number, and the maximum degree of G, respectively. In this paper, we prove that χ_ef(G) = χ_e(G) = Δ(G) for any 2‐connected simple plane graph G with Δ (G) ≥ 24. © 2005 Wiley Periodicals, Inc. J Graph Theory     

On the complexity of the circular chromatic number

Circular chromatic number, χ_c is a natural generalization of chromatic number. It is known that it is NP ‐hard to determine whether or not an arbitrary graph G satisfies χ(G)=χ_c(G). In this paper we prove that this problem is NP ‐hard even if the chromatic number of the graph is known. This answers a question of Xuding Zhu. Also we prove that for all positive integers k ≥ 2 and n ≥ 3, for a given graph G with χ(G) = n, it is NP ‐complete to verify if . © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 226–230, 2004     

Inequalities for the first‐fit chromatic number

The First‐Fit (or Grundy) chromatic number of G, written as χ_FF(G), is defined as the maximum number of classes in an ordered partition of V(G) into independent sets so that each vertex has a neighbor in each set earlier than its own. The well‐known Nordhaus‐‐Gaddum inequality states that the sum of the ordinary chromatic numbers of an n‐vertex graph and its complement is at most n + 1. Zaker suggested finding the analogous inequality for the First‐Fit chromatic number. We show for n ≥ 10 that ?(5n + 2)/4? is an upper bound, and this is sharp. We extend the problem for multicolorings as well and prove asymptotic results for infinitely many cases. We also show that the smallest order of C₄‐free bipartite graphs with χ_FF(G) = k is asymptotically 2k² (the upper bound answers a problem of Zaker [9]). © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 75–88, 2008     

Random lifts of graphs: Independence and chromatic number

For a graph G, a random n‐lift of G has the vertex set V(G)×[n] and for each edge [u, v] ∈ E(G), there is a random matching between {u}×[n] and {v}×[n]. We present bounds on the chromatic number and on the independence number of typical random lifts, with G fixed and n→∞. For the independence number, upper and lower bounds are obtained as solutions to certain optimization problems on the base graph. For a base graph G with chromatic number χ and fractional chromatic number χ_f, we show that the chromatic number of typical lifts is bounded from below by const ? and also by const ? χ_f/log²χ_f (trivially, it is bounded by χ from above). We have examples of graphs where the chromatic number of the lift equals χ almost surely, and others where it is a.s. O(χ/logχ). Many interesting problems remain open. © 2002 John Wiley & Sons, Inc. Random Struct. Alg., 20, 1–22, 2002     

Path chromatic numbers of graphs

Let the finite, simple, undirected graph G = (V(G), E(G)) be vertex-colored. Denote the distinct colors by 1,2,…,c. Let V_i be the set of all vertices colored j and let <V_i be the subgraph of G induced by V_i. The k-path chromatic number of G, denoted by χ(G; P_k), is the least number c of distinct colors with which V(G) can be colored such that each connected component of V_i is a path of order at most k, 1 ? i ? c. We obtain upper bounds for χ(G; P_k) and χ(G; P_∞) for regular, planar, and outerplanar graphs.     

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     

Minimum degree thresholds for bipartite graph tiling

Given a bipartite graph H and a positive integer n such that v(H) divides 2n, we define the minimum degree threshold for bipartite H‐tiling, δ₂(n, H), as the smallest integer k such that every bipartite graph G with n vertices in each partition and minimum degree δ(G)≥k contains a spanning subgraph consisting of vertex‐disjoint copies of H. Zhao, Hladký‐Schacht, Czygrinow‐DeBiasio determined δ₂(n, K_{s, t}) exactly for all s?t and suffi‐ciently large n. In this article we determine δ₂(n, H), up to an additive constant, for all bipartite H and sufficiently large n. Additionally, we give a corresponding minimum degree threshold to guarantee that G has an H‐tiling missing only a constant number of vertices. Our δ₂(n, H) depends on either the chromatic number χ(H) or the critical chromatic number χ_cr(H), while the threshold for the almost perfect tiling only depends on χ_cr(H). These results can be viewed as bipartite analogs to the results of Kuhn and Osthus [Combinatorica 29 (2009), 65–107] and of Shokoufandeh and Zhao [Rand Struc Alg 23 (2003), 180–205]. © 2011 Wiley Periodicals, Inc. J Graph Theory     

A ramsey-theoretic result involving chromatic numbers

The following result is proved. A graph G can be expressed as the edge-disjoint union of k graphs having chromatic numbers no greater than m₁,…,m_k, respectively, iff χ(G) ≤ m₁…m_k.     

Chromatic factorizations of a graph

Let n₁ ? n₂ ? …? ? n_k ? 2 be integers. We say that G has an (n₁, n₂, …?, n_k-chromatic factorization if G) can be edge-factored as G₁ ⊕ G₂ ⊕ …? ⊕ G_k with χ(G_i) = nA_i, for i = 1,2,…, k. The following results are proved:

• i If (n₁ ? 1)n₂ …? n_k < χ(G) ? n₁n₂ …? n_k, then G has an (n₁, n₂, …?, n_k)-chromatic factorization.
• ii If n₁ + n₂ + …? + n_k ? (k - 1) ? n ? n₁n₂ …? n_k, then K_n has an (n₁, n₂, …?, n_k)-chromatic factorization.

     

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     

Equitable coloring of random graphs

An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most one. The least positive integer k for which there exists an equitable coloring of a graph G with k colors is said to be the equitable chromatic number of G and is denoted by χ₌(G). The least positive integer k such that for any k′ ≥ k there exists an equitable coloring of a graph G with k′ colors is said to be the equitable chromatic threshold of G and is denoted by χ₌*(G). In this paper, we investigate the asymptotic behavior of these coloring parameters in the probability space G(n,p) of random graphs. We prove that if n^?1/5+? < p < 0.99 for some 0 < ?, then almost surely χ(G(n,p)) ≤ χ₌(G(n,p)) = (1 + o(1))χ(G(n,p)) holds (where χ(G(n,p)) is the ordinary chromatic number of G(n,p)). We also show that there exists a constant C such that if C/n < p < 0.99, then almost surely χ(G(n,p)) ≤ χ₌(G(n,p)) ≤ (2 + o(1))χ(G(n,p)). Concerning the equitable chromatic threshold, we prove that if n^?(1??) < p < 0.99 for some 0 < ?, then almost surely χ(G(n,p)) ≤ χ₌* (G(n,p)) ≤ (2 + o(1))χ(G(n,p)) holds, and if < p < 0.99 for some 0 < ?, then almost surely we have χ(G(n,p)) ≤ χ₌*(G(n,p)) = O_?(χ(G(n,p))). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

A Remark on Hamiltonian Cycles

Let G be an undirected and simple graph on n vertices. Let ω, α and χ denote the number of components, the independence number and the connectivity number of G. G is called a 1-tough graph if ω(G – S) ? |S| for any subset S of V(G) such that ω(G ? S) > 1. Let σ₂ = min {d(v) + d(w)|v and w are nonadjacent}. Note that the difference α - χ in 1-tough graph may be made arbitrary large. In this paper we prove that any 1-tough graph with σ₂ > n + χ - α is hamiltonian.     

An asymptotically tight bound on the adaptable chromatic number

The adaptable chromatic number of a multigraph G, denoted χ_a(G), is the smallest integer k such that every edge labeling, τ, of G from [k] = {1, 2, …, k} permits a vertex coloring, σ, of G from [k] such that no edge e = uv has τ(e) = σ(u) = σ(v). Hell and Zhu proved that for any multigraph G with maximum degree Δ, the adaptable chromatic number is at most . We strengthen this to the asymptotically best possible bound of for any ?>0. © 2012 Wiley Periodicals, Inc. J Graph Theory     

Claw‐free circular‐perfect graphs

The circular chromatic number of a graph is a well‐studied refinement of the chromatic number. Circular‐perfect graphs form a superclass of perfect graphs defined by means of this more general coloring concept. This article studies claw‐free circular‐perfect graphs. First, we prove that if G is a connected claw‐free circular‐perfect graph with χ(G)>ω(G), then min{α(G), ω(G)}=2. We use this result to design a polynomial time algorithm that computes the circular chromatic number of claw‐free circular‐perfect graphs. A consequence of the strong perfect graph theorem is that minimal imperfect graphs G have min{α(G), ω(G)}=2. In contrast to this result, it is shown in Z. Pan and X. Zhu [European J Combin 29(4) (2008), 1055–1063] that minimal circular‐imperfect graphs G can have arbitrarily large independence number and arbitrarily large clique number. In this article, we prove that claw‐free minimal circular‐imperfect graphs G have min{α(G), ω(G)}≤3. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 163–172, 2010     

Cochromatic Number and the Genus of a Graph

The cochromatic number of a graph G, denoted by z(G), is the minimum number of subsets into which the vertex set of G can be partitioned so that each sbuset induces an empty or a complete subgraph of G. In this paper we introduce the problem of determining for a surface S, z(S), which is the maximum cochromatic number among all graphs G that embed in S. Some general bounds are obtained; for example, it is shown that if S is orientable of genus at least one, or if S is nonorientable of genus at least four, then z(S) is nonorientable of genus at least four, then z(S)≤χ(S). Here χ(S) denotes the chromatic number S. Exact results are obtained for the sphere, the Klein bottle, and for S. It is conjectured that z(S) is equal to the maximum n for which the graph G_n = K₁ ∪ K₂ ∪ … ∪ K_n embeds in S.     

Partitioning complete multipartite graphs by monochromatic trees

The tree partition number of an r‐edge‐colored graph G, denoted by t_r(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex‐disjoint monochromatic trees. We determine t₂(K(n₁, n₂,…, n_k)) of the complete k‐partite graph K(n₁, n₂,…, n_k). In particular, we prove that t₂(K(n, m)) = ? (m‐2)/2ⁿ? + 2, where 1 ≤ n ≤ m. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 133–141, 2005     

Size ramsey numbers of stars versus 4‐chromatic graphs

We investigate the asymptotics of the size Ramsey number î(K_1,nF), where K_1,n is the n‐star and F is a fixed graph. The author 11 has recently proved that r?(K_1,n,F)=(1+o(1))n² for any F with chromatic number χ(F)=3. Here we show that r?(K_1,n,F)≤ n²+o(n²), if χ (F) ≥ 4 and conjecture that this is sharp. We prove the case χ(F)=4 of the conjecture, that is, that r?(K_1,n,F)=(4+o(1))n² for any 4‐chromatic graph F. Also, some general lower bounds are obtained. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 220–233, 2003     

On Rosa-type labelings and cyclic graph decompositions

A labeling (or valuation) of a graph G is an assignment of integers to the vertices of G subject to certain conditions. A hierarchy of graph labelings was introduced by Rosa in the late 1960s. Rosa showed that certain basic labelings of a graph G with n edges yielded cyclic G-decompositions of K _2n+1 while other stricter labelings yielded cyclic G-decompositions of K _2nx+1 for all natural numbers x. Rosa-type labelings are labelings with applications to cyclic graph decompositions. We survey various Rosa-type labelings and summarize some of the related results. (Communicated by Peter Horák)