The chromatic polynomials of some families of quadrangulations of the torus can be found explicitly. The method, known as ‘bracelet theory’ is based on a decomposition in terms of representations of the symmetric group. The results are particularly appropriate for studying the limit curves of the chromatic roots of these families. In this paper these techniques are applied to a family of quadra...

It is well-known (Feige and Kilian [24], H̊astad [39]) that approximating the chromatic number within a factor of n1−ε cannot be done in polynomial time for ε > 0, unless coRP = NP. Computing the list-chromatic number is much harder than determining the chromatic number. It is known that the problem of deciding if the list-chromatic number is k, where k ≥ 3, is Πp2-complete [37]. In this paper, ...

A graph G is chordless if no cycle in G has a chord. In the present work we investigate the chromatic index and total chromatic number of chordless graphs. We describe a known decomposition result for chordless graphs and use it to establish that every chordless graph of maximum degree ∆ ≥ 3 has chromatic index ∆ and total chromatic number ∆+1. The proofs are algorithmic in the sense that we ac...

We study the probability that a random polygon of k vertices in the lattice {1, . . . , n}s does not contain more lattice points than the k vertices of the polygon. Then we introduce the chromatic zeta function of a graph to generalize this problem to other configurations induced by a given graph H.

We develop a new upper bound, called the nested chromatic number, for the chromatic number of a finite simple graph. This new invariant can be computed in polynomial time, unlike the standard chromatic number which is NP -hard. We further develop multiple distinct bounds on the nested chromatic number using common properties of graphs. We also determine the behavior of the nested chromatic numb...

In this paper we prove the knight move theorem for the chromatic graph cohomologies with rational coefficients introduced by L. HelmeGuizon and Y. Rong. Namely, for a connected graph Γ with n vertices the only non-trivial cohomology groups H(Γ), H(Γ) come in isomorphic pairs: H(Γ) = H(Γ) for i > 0 if Γ is non-bipartite, and for i > 0 if Γ is bipartite. As a corollary, the ranks of the cohomolog...

A main result of combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time (Grötschel, Lovász and Schrijver 1981). Circular-perfect graphs form a well-studied superclass of perfect graphs. We extend the above result for perfect graphs by showing that clique and chromatic number of a circularperfect graph are computable in polynomial time...

1 1 Introduction 2 1 Introduction Suppose Γ is a graph with |V (Γ)| = n. For λ a positive integer, let [λ] = {1, 2,. .. , λ} be a set of λ distinct colors. A λ-coloring of Γ is a mapping f from V (Γ) to [λ]. Whenever for every two adjacent vertices u and v, f (u) = f (v), we will call f a proper coloring of Γ; otherwise, improper. When a proper λ-coloring exists, we call Γ a λ-colorable graph. ...

J. Makowsky and B. Zilber (2004) showed that many variations of graph colorings, called CP-colorings in the sequel, give rise to graph polynomials. This is true in particular for harmonious colorings, convex colorings, mcct-colorings, and rainbow colorings, and many more. N. Linial (1986) showed that the chromatic polynomial χ(G;X) is #P-hard to evaluate for all but three values X = 0, 1, 2, wh...