Related papers: Generalized chromatic functions
An $r$-edge coloring of a graph or hypergraph $G=(V,E)$ is a map $c:E\to \{0, \dots, r-1\}$. Extending results of Rado and answering questions of Rado, Gy\'arf\'as and S\'ark\"ozy we prove that (1.) the vertex set of every $r$-edge colored…
In this paper we consider a colouring version of the general position problem. The \emph{$\gp $-chromatic number} is the smallest number of colours needed to colour the vertices of the graph such that each colour class has the…
A complete partition of a graph $G$ is a partition of the vertex set such that there is at least one edge between any two parts. The largest $r$ such that $G$ has a complete partition into $r$ parts, each of which is an independent set, is…
We introduce a generalized Grover matrix of a graph and present an explicit formula for its characteristic polynomial. As a corollary, we give the spectra for the generalized Grover matrix of a regular graph. Next, we define a zeta function…
In this paper we introduce the notion of $\Sigma$-colouring of a graph $G$: For given subsets $\Sigma(v)$ of neighbours of $v$, for every $v\in V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that…
The representation is essentially the same as that given by J.P.Nagle in J. Comb. Theory (B), 1971, 10:1, 42--59. The distinction is in the definition of the weighting function via the number of flows. This new definition allows one to…
Following Penrose, we introduce a family of graph functions defined in terms of contractions of certain products of symmetric tensors along the edges of a graph. Special cases of these functions enumerate edge colorings and cycles of…
DP-coloring is a generalization of list coloring that was introduced in 2015 by Dvo\v{r}\'{a}k and Postle. The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$. A well-known tool…
Erd\H{o}s proved that there are graphs with arbitrarily large girth and chromatic number. We study the extension of this for generalized chromatic numbers.
It is proved that 1) the indicator function of some onefold or multifold independent set in a regular graph is a perfect coloring if and only if the set attain the Delsarte--Hoffman bound; 2) each transversal in a uniform regular hypergraph…
Colouring the vertices of a graph $G$ according to certain conditions can be considered as a random experiment and a discrete random variable $X$ can be defined as the number of vertices having a particular colour in the proper colouring of…
We introduce a quasisymmetric class function associated with a group acting on a double poset or on a directed graph. The latter is a generalization of the chromatic quasisymmetric function of a digraph introduced by Ellzey, while the…
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…
Asymptotic expansions of Gaussian integrals may often be interpreted as generating functions for certain combinatorial objects (graphs with additional data). In this article we discuss a general approach to all such cases using colored…
Many variations of the classical graph coloring model have been intensively studied due to their multiple applications; scheduling problems and aircraft assignments, for instance, motivate the robust coloring problem. This model gets to…
An edge-coloring of a graph $G$ with natural numbers is called a sum edge-coloring if the colors of edges incident to any vertex of $G$ are distinct and the sum of the colors of the edges of $G$ is minimum. The edge-chromatic sum of a graph…
Let $\CRF_S$ denote the category of $S$-colored rooted forests, and $\H_{\CRF_S}$ denote its Ringel-Hall algebra as introduced in \cite{KS}. We construct a homomorphism from a $K^+_0 (\CRF_S)$--graded version of the Hopf algebra of…
One of the fundamental and most-studied algorithmic problems in distributed computing on networks is graph coloring, both in bounded-degree and in general graphs. Recently, the study of this problem has been extended in two directions.…
The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as…
We study several basic problems about colouring the $p$-random subgraph $G_p$ of an arbitrary graph $G$, focusing primarily on the chromatic number and colouring number of $G_p$. In particular, we show that there exist infinitely many…