Optimization of eigenvalue bounds for the independence and chromatic number of graph powers

The $k^{\text{th}}$ power of a graph $G=(V,E)$, $G^k$, is the graph whose vertex set is V and in which two distinct vertices are adjacent if and only if their distance in G is at most k. This article proves various eigenvalue bounds for the independence number and chromatic number of $G^k$ which purely depend on the spectrum of G, together with a method to optimize them. Our bounds for the k-independence number also work for its quantum counterpart, which is not known to be a computable parameter in general, thus justifying the use of integer programming to optimize them. Some of the bounds previously known in the literature follow as a corollary of our main results. Infinite families of graphs where the bounds are sharp are presented as well.     

New bounds for the chromatic number of graphs

In this article we first give an upper bound for the chromatic number of a graph in terms of its degrees. This bound generalizes and modifies the bound given in 11 . Next, we obtain an upper bound of the order of magnitude for the coloring number of a graph with small K_2,t (as subgraph), where n is the order of the graph. Finally, we give some bounds for chromatic number in terms of girth and book size. These bounds improve the best known bound, in terms of order and girth, for the chromatic number of a graph when its girth is an even integer. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:110–122, 2008     

Spectral bounds for the clique and independence numbers of graphs

We obtain a sequence k₁(G) ≤ k₂(G) ≤ … ≤ k_n(G) of lower bounds for the clique number (size of the largest clique) of a graph G of n vertices. The bounds involve the spectrum of the adjacency matrix of G. The bound k₁(G) is explicit and improves earlier known theorems. The bound k₂(G) is also explicit, and is shown to improve on the bound from Brooks' theorem even for regular graphs. The bounds k₃,…, k_r are polynomial-time computable, where r is the number of positive eigenvalues of G.     

The independence number of an edge‐chromatic critical graph

A graph G with maximum degree Δ and edge chromatic number χ′(G)>Δ is edge‐Δ‐critical if χ′(G?e)=Δ for every edge e of G. It is proved here that the vertex independence number of an edge‐Δ‐critical graph of order n is less than . For large Δ, this improves on the best bound previously known, which was roughly ; the bound conjectured by Vizing, which would be best possible, is . © 2010 Wiley Periodicals, Inc. J Graph Theory 66:98‐103, 2011     

Three new upper bounds on the chromatic number

This paper introduces three new upper bounds on the chromatic number, without making any assumptions on the graph structure. The first one, ξ, is based on the number of edges and nodes, and is to be applied to any connected component of the graph, whereas ζ and η are based on the degree of the nodes in the graph. The computation complexity of the three-bound computation is assessed. Theoretical and computational comparisons are also made with five well-known bounds from the literature, which demonstrate the superiority of the new upper bounds.     

Improved bounds on the generalized acyclic chromatic number

An r-acyclic edge chromatic number of a graph G, denoted by a _r ^′ r(G), is the minimum number of colors used to produce an edge coloring of the graph such that adjacent edges receive different colors and every cycle C has at least min {|C|, r} colors. We prove that a _r ^′ r(G) ≤ (4r + 1)Δ(G), when the girth of the graph G equals to max{50, Δ(G)} and 4 ≤ r ≤ 7. If we relax the restriction of the girth to max {220, Δ(G)}, the upper bound of a _r ^′ r(G) is not larger than (2r + 5)Δ(G) with 4 ≤ r ≤ 10.     

New upper bounds for the chromatic number of a graph*

We show that for any graph G, the chromatic number χ(G) ≤ Δ₂(G) + 1, where Δ₂(G) is the largest degree that a vertex ν can have subject to the condition that ν is adjacent to a vertex whose degree is at least as big as its own. Moreover, we show that the upper bound is best possible in the the following sense: If Δ₂(G) ≥ 3, then to determine whether χ(G) ≤ Δ₂(G) is an NP‐complete problem. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 117–120, 2001     

Some bounds on the injective chromatic number of graphs

We give some bounds on the injective chromatic number.     

Upper bounds on the independence and the clique covering number

New upper bounds for the independence number and for the clique covering number of a graph are given in terms of the rank, respectively the eigenvalues, of the adjacency matrix. We formulate a conjecture concerning an upper bound of the clique covering number. This upper bound is related to an old conjecture of Alan J. Hoffman which is shown to be false. Key words: adjacency matrix, eigenvalues, independence number, clique covering number. AMS classification: 05C.     

On bounds for the incidentor chromatic number of a directed weighted multigraph

An incidentor coloring of a directed weighted multigraph is called admissible if: (a) the incidentors adjoining the same vertex are colored by different colors; (b) the difference between the colors of the final and initial incidentors of each arc is at least the weight of this arc. The minimum number of colors necessary for an admissible coloring of all incidentors of a multigraph G is bounded above and below. The upper and lower bounds differ by ┌Δ/2┐ where Δ is the degree of G.     

On lower bounds for the chromatic number in terms of vertex degree

In this paper we discuss the existence of lower bounds for the chromatic number of graphs in terms of the average degree or the coloring number of graphs. We obtain a lower bound for the chromatic number of K_1,t-free graphs in terms of the maximum degree and show that the bound is tight. For any tree T, we obtain a lower bound for the chromatic number of any K_2,t-free and T-free graph in terms of its average degree. This answers affirmatively a modified version of Problem 4.3 in [T.R. Jensen, B. Toft, Graph Coloring Problems, Wiley, New York, 1995]. More generally, we discuss δ-bounded families of graphs and then we obtain a necessary and sufficient condition for a family of graphs to be a δ-bounded family in terms of its induced bipartite Turán number. Our last bound is in terms of forbidden induced even cycles in graphs; it extends a result in [S.E. Markossian, G.S. Gasparian, B.A. Reed, β-perfect graphs, J. Combin. Theory Ser. B 67 (1996) 1–11].     

Upper bounds on the linear chromatic number of a graph

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is a union of vertex-disjoint paths. The linear chromatic number of G is the smallest number of colors in a linear coloring of G.Let G be a graph with maximum degree Δ(G). In this paper we prove the following results: (1) ; (2) if Δ(G)≤4; (3) if Δ(G)≤5; (4) if G is planar and Δ(G)≥52.     

A new upper bound for the independence number of edge chromatic critical graphs

In 1968, Vizing conjectured that if G is a Δ‐critical graph with n vertices, then α(G)≤n/2, where α(G) is the independence number of G. In this paper, we apply Vizing and Vizing‐like adjacency lemmas to this problem and prove that α(G)<(((5Δ?6)n)/(8Δ?6))<5n/8 if Δ≥6. © 2010 Wiley Periodicals, Inc. J Graph Theory 68: 202‐212, 2011     

图的邻点可区别全色数的一个上界

图G的一个正常全染色被称为邻点可区别全染色,如果G中任意两个相邻点的色集合不同．本文用概率方法得到了邻点可区别全色数的一个上界．     

Spectral bounds of the difference Laplace operator in non-rectangular regions

We consider the eigenvalue problem for a two-dimensional difference Laplace operator in non-rectangular regions (a curvilinear triangle, a curvilinear trapezoid, a circular segment). The dependence of the eigenvalues on the parameters of the regions is elucidated. The main result is the derivation of the spectral bounds of the difference operator. A lower bound for the minimum eigenvalue and an upper bound for the maximum eigenvalue are determined. The spectral bound is determined numerically for a series of non-rectangular regions. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 94–113, 2006.