Related papers: Harary polynomials

DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvo\v{r}\'{a}k and Postle in 2015. As the analogue of the chromatic polynomial of…

We show that every NP problem is polynomially equivalent to a simple combinatorial problem: the membership problem for a special class of digraphs. These classes are defined by means of shadows (projections) and by finitely many forbidden…

For a bipartite graph $G$ with parts $X$ and $Y$, an $X$-interval coloring is a proper edge coloring of $G$ by integers such that the colors on the edges incident to any vertex in $X$ form an interval. Denote by $\chi'_{int}(G,X)$ the…

We study a $q$-version of the chromatic polynomial of a given graph $G=(V,E)$, namely, \[ \chi_G^\lambda(q,n) \ := \sum_{\substack{\text{proper colorings}\\ c\,:\,V\to[n]}} q^{ \sum_{ v \in V } \lambda_v c(v) }, \] where $\lambda \in…

For a graph $F$, a graph $G$ is \emph{$F$-free} if it does not contain an induced subgraph isomorphic to $F$. For two graphs $G$ and $H$, an \emph{$H$-coloring} of $G$ is a mapping $f:V(G)\rightarrow V(H)$ such that for every edge $uv\in…

This paper studies some coloring properties of graph powers. We show that $\chi_c(G^{^{\frac{2r+1}{2s+1}}})=\frac{(2s+1)\chi_c(G)}{(s-r)\chi_c(G)+2r+1}$ provided that $\chi_c(G^{^{\frac{2r+1}{2s+1}}})< 4$. As a consequence, one can see that…

We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction…

Fix $d\ge2$ and consider a uniformly random set $P$ of $n$ points in $[0,1]^{d}$. Let $G$ be the Hasse diagram of $P$ (with respect to the coordinatewise partial order), or alternatively let $G$ be the Delaunay graph of $P$ with respect to…

We introduce ideas that complement the many known connections between polymatroids and graph coloring. Given a hypergraph that satisfies certain conditions, we construct polymatroids, given as rank functions, that can be written as sums of…

There are many variations on partition functions for graph homomorphisms or colorings. The case considered here is a counting or hard constraint problem in which the range or color graph carries a free and vertex transitive Abelian group…

It is proved that every connected graph $G$ on $n$ vertices with $\chi(G) \geq 4$ has at most $k(k-1)^{n-3}(k-2)(k-3)$ $k$-colourings for every $k \geq 4$. Equality holds for some (and then for every) $k$ if and only if the graph is formed…

A hole in a graph $G$ is an induced cycle of length at least four, and a $k$-multihole in $G$ is a set of pairwise disjoint and nonadjacent holes. It is well known that if $G$ does not contain any holes then its chromatic number is equal to…

In this article we describe a new inductive approach to compute the chromatic polynomial of simple graphs and the characteristic polynomial of central hyperplane arrangements.

DP-coloring (also called correspondence coloring) is a well-studied generalization of list coloring introduced by Dvo\v{r}\'{a}k and Postle in 2015. The following sharp bound on the DP-chromatic number of the Cartesian product of graphs $G$…

In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and…

Let $G$ be a bipartite graph with adjacency matrix $A(G)$. The characteristic polynomial $\phi(G,x)=\det(xI-A(G))$ and the permanental polynomial $\pi(G,x) = \text{per}(xI-A(G))$ are both graph invariants used to distinguish graphs. For…

We introduce a class of pairs of graphs consisting of two cliques joined by an arbitrary number of edges. The members of a pair have the property that the clique-bridging edge-set of one graph is the complement of that of the other. We…

DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvo\v{r}\'{a}k and Postle. In this paper we introduce and study the fractional DP-chromatic number $\chi_{DP}^\ast(G)$. We…

For a graph $H$ and integer $k \geq 1$, two functions $f, g$ from $V(H)$ into $\{1, \dots, k\}$ are adjacent if for all edges $uv$ of $H$, $f(u) \neq g(v)$. The graph of all such functions is the exponential graph $K_k^H$. El-Zahar and…

Harary et al. and Klein and Randic proposed the forcing number of a perfect matching in mathematics and chemistry, respectively. In detail, the forcing number of a perfect matching M of a graph G is the smallest cardinality of subsets of M…