The total chromatic number of regular graphs of high degree

The total chromatic number χT (G) of a graph G is the minimum number of colors needed to color the edges and the vertices of G so that incident or adjacent elements have distinct colors. We show that if G is a regular graph and d(G) 32 |V (G)| + 263 , where d(G) denotes the degree of a vertex in G, then χT (G) d(G) + 2.     

Building fences around the chromatic coefficients

Associated to each graph G is its chromatic polynomial f(G, t) and we associate to f(G, t) the sequence α (G) of the norms of its coefficients. A stringent partial ordering is established for such sequences. The main result is that for any graph G with q edges we have α (R_q) ≤ α (G) ≤ α (S_q), where R_q and S_q are specified graphs with q edges. This translates into a clearer view of allowable values and patterns in the chromatic coefficients. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 123–128, 1997     

Some Algebraic Methods for Calculating the Number of Colorings of a Graph

With an arbitrary graph G having n vertices and m edges, and with an arbitrary natural number p, we associate in a natural way a polynomial R(x ₁,...,x _n) with integer coefficients such that the number of colorings of the vertices of the graph G in p colors is equal to p ^m-n R(0,...,0). Also with an arbitrary maximal planar graph G, we associate several polynomials with integer coefficients such that the number of colorings of the edges of the graph G in 3 colors can be calculated in several ways via the coefficients of each of these polynomials. Bibliography: 2 titles.     

On a conjecture by Plummer and Toft

The cyclic chromatic number χ_c(G) of a 2‐connected plane graph G is the minimum number of colors in an assigment of colors to the vertices of G such that, for every face‐bounding cycle f of G, the vertices of f have different colors. Plummer and Toft proved that, for a 3‐connected plane graph G, under the assumption Δ*(G) ≥ 42, where Δ*(G) is the size of a largest face of G, it holds that χ_c(G) ≤ Δ*(G) + 4. They conjectured that, if G is a 3‐connected plane graph, then χ_c>(G) ≤ Δ*(G) + 2. In the article the conjecture is proved for Δ*(G) ≥ 24. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 177–189, 1999     

There exist graphs with super‐exponential Ramsey multiplicity constant

The Ramsey multiplicity M(G;n) of a graph G is the minimum number of monochromatic copies of G over all 2‐colorings of the edges of the complete graph K_n. For a graph G with a automorphisms, ν vertices, and E edges, it is natural to define the Ramsey multiplicity constant C(G) to be , which is the limit of the fraction of the total number of copies of G which must be monochromatic in a 2‐coloring of the edges of K_n. In 1980, Burr and Rosta showed that 0 ≥ C(G) ≤ 2^1?E for all graphs G, and conjectured that this upper bound is tight. Counterexamples of Burr and Rosta's conjecture were first found by Sidorenko and Thomason independently. Later, Clark proved that there are graphs G with E edges and 2^E?1C(G) arbitrarily small. We prove that for each positive integer E, there is a graph G with E edges and C(G) ≤ E^{?E/2 + o(E)}. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 89–98, 2008     

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     

List circular coloring of trees and cycles

Suppose G=(V, E) is a graph and p ≥ 2q are positive integers. A (p, q)‐coloring of G is a mapping ?: V → {0, 1, …, p‐1} such that for any edge xy of G, q ≤ |?(x)‐?(y)| ≤ p‐q. A color‐list is a mapping L: V → ({0, 1, …, p‐1}) which assigns to each vertex v a set L(v) of permissible colors. An L‐(p, q)‐coloring of G is a (p, q)‐coloring ? of G such that for each vertex v, ?(v) ∈ L(v). We say G is L‐(p, q)‐colorable if there exists an L‐(p, q)‐coloring of G. A color‐size‐list is a mapping ? which assigns to each vertex v a non‐negative integer ?(v). We say G is ?‐(p, q)‐colorable if for every color‐list L with |L(v)| = ?(v), G is L‐(p, q)‐colorable. In this article, we consider list circular coloring of trees and cycles. For any tree T and for any p ≥ 2q, we present a necessary and sufficient condition for T to be ?‐(p, q)‐colorable. For each cycle C and for each positive integer k, we present a condition on ? which is sufficient for C to be ?‐(2k+1, k)‐colorable, and the condition is sharp. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 249–265, 2007     

T-Colorings and T-Edge Spans of Graphs

 Suppose G is a graph and T is a set of non-negative integers that contains 0. A T-coloring of G is an assignment of a non-negative integer f(x) to each vertex x of G such that |f(x)−f(y)|∉T whenever xy∈E(G). The edge span of a T-coloring−f is the maximum value of |f(x) f(y)| over all edges xy, and the T-edge span of a graph G is the minimum value of the edge span of a T-coloring of G. This paper studies the T-edge span of the dth power C ^d _n of the n-cycle C _n for T={0, 1, 2, …, k−1}. In particular, we find the exact value of the T-edge span of C _n ^d for n≡0 or (mod d+1), and lower and upper bounds for other cases. Received: May 13, 1996 Revised: December 8, 1997     

Sharp bounds for the number of 3‐independent partitions and the chromaticity of bipartite graphs

Given a graph G and an integer k ≥ 1, let α(G, k) denote the number of k‐independent partitions of G. Let ??^?s(p,q) (resp., ??₂^?s(p,q)) denote the family of connected (resp., 2‐connected) graphs which are obtained from the complete bipartite graph K_p,q by deleting a set of s edges, where p ≥ q ≥ 2. This paper first gives a sharp upper bound for α(G,3), where G ∈ ?? ^?s(p,q) and 0 ≤ s ≤ (p ? 1)(q ? 1) (resp., G ∈ ?? ₂^?s(p,q) and 0 ≤ s ≤ p + q ? 4). These bounds are then used to show that if G ∈ ?? ^?s(p,q) (resp., G ∈ ?? ₂^?s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets ??^?s_i(p+i,q?i) where max and s_i = s ? i(p?q+i) (resp., a subset of ??₂^?s(p,q), where either 0 ≤ s ≤ q ? 1, or s ≤ 2q ? 3 and p ≥ q + 4). By applying these results, we show finally that any 2‐connected graph obtained from K_p,q by deleting a set of edges that forms a matching of size at most q ? 1 or that induces a star is chromatically unique. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 48–77, 2001     

On totally multicolored stars

Given an edge‐coloring of a graph G, a subgraph M of G will be called totally multicolored if no two edges of M receive the same color. Let h(G, K_1,q) be the minimum integer such that every edge‐coloring of G using exactly h(G, K_1,q) colors produces at least one totally multicolored copy of K_1,q (the q‐star) in G. In this article, an upper bound of h(G, K_1,q) is presented, as well as some applications of this upper bound. © 2005 Wiley Periodicals, Inc.     

Strong chromatic index of subset graphs

The strong chromatic index of a graph G, denoted sq(G), is the minimum number of parts needed to partition the edges of G into induced matchings. For 0 ≤ k ≤ l ≤ m, the subset graph S_m(k, l) is a bipartite graph whose vertices are the k- and l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. We show that and that this number satisfies the strong chromatic index conjecture by Brualdi and Quinn for bipartite graphs. Further, we demonstrate that the conjecture is also valid for a more general family of bipartite graphs. © 1997 John Wiley & Sons, Inc.     

Irredundance in inflated graphs

The inflation G_I of a graph G with n(G) vertices and m(G) edges is obtained by replacing every vertex of degree d of G by a clique K_d. We study the lower and upper irredundance parameters ir and IR of an inflation. We prove in particular that if γ denotes the domination number of a graph, γ(G_I) − ir(G_I) can be arbitrarily large, IR(G_I) ≤ m(G) and IR(G_I) ≤ n²(G)/4. These results disprove a conjecture of Dunbar and Haynes (Congr. Num. 118 (1996), 143–154) and answer another open question. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 97–104, 1998     

Total chromatic number of one kind of join graphs

The total chromatic number χ_T(G) of a graph G is the minimum number of colors needed to color the elements (vertices and edges) of G such that no adjacent or incident pair of elements receive the same color. G is called Type 1 if χ_T(G)=Δ(G)+1. In this paper we prove that the join of a complete inequibipartite graph K_n1,n2 and a path P_m is of Type 1.     

Some extremal results in cochromatic and dichromatic theory

For a graph G, the cochromatic number of G, denoted z(G), is the least m for which there is a partition of the vertex set of G having order m. where each part induces a complete or empty graph. We show that if {G_n} is a family of graphs where G_n has o(n² log²(n)) edges, then z(G_n) = o(n). We turn our attention to dichromatic numbers. Given a digraph D, the dichromatic number of D is the minimum number of parts the vertex set of D must be partitioned into so that each part induces an acyclic digraph. Given an (undirected) graph G, the dichromatic number of G, denoted d(G), is the maximum dichromatic number of all orientations of G. Let m be an integer; by d(m) we mean the minimum size of all graphs G where d(G) = m. We show that d(m) = θ(m² ln²(m)).     

The total chromatic number of Pseudo-Halin graphs with lower degree

The total chromatic number χ_T(G) of a graph G is the least number of colors needed to color the vertices and the edges of G such that no adjacent or incident elements receive the same color. The Total Coloring Conjecture(TCC) states that for every simple graph G, χ_T(G)≤Δ(G)+2. In this paper, we show that χ_T(G)=Δ(G)+1 for all pseudo-Halin graphs with Δ(G)=4 and 5.     

Acyclic edge colorings of graphs

A proper coloring of the edges of a graph G is called acyclic if there is no 2‐colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a′(G) ≥ Δ(G) + 2 where Δ(G) is the maximum degree in G. It is known that a′(G) ≤ 16 Δ(G) for any graph G. We prove that there exists a constant c such that a′(G) ≤ Δ(G) + 2 for any graph G whose girth is at least cΔ(G) log Δ(G), and conjecture that this upper bound for a′(G) holds for all graphs G. We also show that a′(G) ≤ Δ + 2 for almost all Δ‐regular graphs. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 157–167, 2001     

Lower bounds for lower Ramsey numbers

For any graph G, let i(G) and μ;(G) denote the smallest number of vertices in a maximal independent set and maximal clique, respectively. For positive integers m and n, the lower Ramsey number s(m, n) is the largest integer p so that every graph of order p has i(G) ≤ m or μ;(G) ≤ n. In this paper we give several new lower bounds for s (m, n) as well as determine precisely the values s(1, n).     

Distinguishing Cartesian powers of graphs

The distinguishing number D(G) of a graph is the least integer d such that there is a d‐labeling of the vertices of G that is not preserved by any nontrivial automorphism of G. We show that the distinguishing number of the square and higher powers of a connected graph G ≠ K₂, K₃ with respect to the Cartesian product is 2. This result strengthens results of Albertson [Electron J Combin, 12 ( 1 ), #N17] on powers of prime graphs, and results of Klav?ar and Zhu [Eu J Combin, to appear]. More generally, we also prove that d(G □ H) = 2 if G and H are relatively prime and |H| ≤ |G| < 2^|H| ? |H|. Under additional conditions similar results hold for powers of graphs with respect to the strong and the direct product. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 250–260, 2006     

[a,b]-factorization of a graph

Let a and b be integers such that 0 ? a ? b. Then a graph G is called an [a, b]-graph if a ? d_G(x) ? b for every x ? V(G), and an [a, b]-factor of a graph is defined to be its spanning subgraph F such that a ? d_F(x) ? b for every vertex x, where d_G(x) and d_F(x) denote the degrees of x in G and F, respectively. If the edges of a graph can be decomposed into [a.b]-factors then we say that the graph is [2a, 2a]-factorable. We prove the following two theorems: (i) a graph G is [2a, 2b)-factorable if and only if G is a [2am,2bm]-graph for some integer m, and (ii) every [8m + 2k, 10m + 2k]-graph is [1,2]-factorable.     

The average Steiner distance of a graph

The average distance μ(G) of a graph G is the average among the distances between all pairs of vertices in G. For n ≥ 2, the average Steiner n-distance μ_n(G) of a connected graph G is the average Steiner distance over all sets of n vertices in G. It is shown that for a connected weighted graph G, μ_n(G) ≤ μ_k(G) + μ_n+1−k(G) where 2 ≤ k ≤ n − 1. The range for the average Steiner n-distance of a connected graph G in terms of n and |V(G)| is established. Moreover, for a tree T and integer k, 2 ≤ k ≤ n − 1, it is shown that μ_n(T) ≤ (n/k)μ_k(T) and the range for μ_n(T) in terms of n and |V(T)| is established. Two efficient algorithms for finding the average Steiner n-distance of a tree are outlined. © 1996 John Wiley & Sons, Inc.