DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvořák and Postle 2015. As the analogue chromatic polynomial P(G,m), DP color function graph G, denoted PDP(G,m), counts minimum number DP-colorings over all possible m-fold covers. Chromatic polynomials for joins vertex-gluings graphs are...

We outline an approach to enumeration problems which relies on the algebra of free abelian groups, giving as our main application a generalisa-tion of the chromatic polynomial of a simple graph G. Our polynomial lies in the free abelian group generated by the poset K(G) of contractions of G, and reduces to the classical case after a simple substitution. Its main properties may be stated in term...

The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobás–Riordan– Tutte polynomial generalizes the Tutte polynomial of graphs to graphs that are embedded in closed oriented surfaces of higher genus. In this paper we show that the Jones polynomial of any lin...

In this note we study a certain graph polynomial arising from a special recursion. This recursion is a member of a family of four recursions where the other three recursions belong to the chromatic polynomial, the modified matching polynomial, and the adjoint polynomial, respectively. The four polynomials share many common properties, for instance all of them are of exponential type, i. e., the...

For a century ago, one of the most famous problems in mathematics was to prove the FourColor Problem. During the period that the Four-Color Problem was unsolved, which spanned more than a century, many approaches were introduced with the hopes that they would lead to a solution of this famous problem. In 1913, Birkhoff 1 defined a function P M,λ that gives the number of proper λ-colorings of a ...

Roots of graph polynomials such as the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we examine to what extent the location of these roots refl ects the graph theoretic properties of the underlying graph. (Joint work with E. Ravve and N. Blanchard) Johann (János) A. Makowsky is a Hungarian born and naturalized Swi...

Recently, Tittmann et al. introduced the subgraph component polynomial and showed that its power for distinguishing graphs is quite different from the power of other graph polynomials that appear in the literature such as the matching polynomial, the Tutte polynomial, the characteristic polynomial, the chromatic polynomial, etc. The subgraph component polynomial enumerates vertex induced subgra...

The visibility graph V(P ) of a point set P ⊆ R has vertex set P , such that two points v, w ∈ P are adjacent whenever there is no other point in P on the line segment between v and w. We study the chromatic number of V(P ). We characterise the 2and 3-chromatic visibility graphs. It is an open problem whether the chromatic number of a visibility graph is bounded by its clique number. Our main r...

A subspace arrangement defined by intersections of hyperplanes of the braid arrangement can be encoded by an edge colored hypergraph. It turns out that the characteristic polynomial of this type of subspace arrangement is given by a generalized chromatic polynomial of the associated edge colored hypergraph. The main result of this paper supplies a sufficient condition for the existence of non-t...

Abstract. We define a q-chromatic function on graphs, list some of its properties and provide some formulas in the class of general chordal graphs. Then we relate the q-chromatic function to the colored Jones function of knots. This leads to a curious expression of the colored Jones function of a knot diagram K as a ’defected chromatic operator’ applied to a power series whose coefficients are ...