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Chromatic polynomials have been studied extensively, giving us results such as the Fundamental Reduction Theorem and closed formulas for the chromatic polynomials of common classes of graphs. Though, none of those extend to the context of…

We present new results on approximate colourings of graphs and, more generally, approximate H-colourings and promise constraint satisfaction problems. First, we show NP-hardness of colouring $k$-colourable graphs with $\binom{k}{\lfloor…

A fractional colouring of a graph $G$ is a function that assigns a non-negative real value to all possible colour-classes of $G$ containing any vertex of $G$, such that the sum of these values is at least one for each vertex. The fractional…

A $k$-coloring of a graph $G=(V,E)$ is called semi-equitable if there exists a partition of its vertex set into independent subsets $V_1,\ldots,V_k$ in such a way that $|V_1| \notin \{\lceil |V|/k\rceil, \lfloor |V|/k \rfloor\}$ and…

A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by \chi_b(G), is the maximum number t such that G admits a…

An $(m,n)$-colored mixed graph $G$ is a graph with its arcs having one of the $m$ different colors and edges having one of the $n$ different colors. A homomorphism $f$ of an $(m,n)$-colored mixed graph $G$ to an $(m,n)$-colored mixed graph…

For integers r and k > 0(k>r),a conditional (k, r)-coloring of a graph G is a proper k-coloring of G such that every vertex v of G has at least min{r,d(v)} differently colored neighbors, where d(v) is the degree of v. In this note, for…

A 2-hued coloring of a graph $G$ (also known as conditional $(k, 2)$-coloring and dynamic coloring) is a coloring such that for every vertex $v\in V(G)$ of degree at least $2$, the neighbors of $v$ receive at least $2$ colors. The smallest…

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…

Let $G$ be a simple connected graph of order $n$. A hamiltonian coloring $c$ of a graph $G$ is an assignment of colors (non-negative integers) to the vertices of $G$ such that $D(u, v)$ + $|c(u) - c(v)|$ $\geq$ $n - 1$ for every two…

We look at colourings of $r$-uniform hypergraphs, focusing our attention on unique colourability and gaps in the chromatic spectrum. The pattern of an edge $E$ in an $r$-uniform hypergraph $H$ whose vertices are coloured is the partition of…

Coloring graphs is an important algorithmic problem in combinatorics with many applications in computer science. In this paper we study coloring tournaments. A chromatic number of a random tournament is of order $\Omega(\frac{n}{\log(n)})$.…

A subcoloring of a graph is a partition of its vertex set into subsets (called colors), each inducing a disjoint union of cliques. It is a natural generalization of the classical proper coloring, in which each color must instead induce an…

For any two non-negative integers h and k, h > k, an L(h, k)-colouring of a graph G is a colouring of vertices such that adjacent vertices admit colours that at least differ by h and vertices that are two distances apart admit colours that…

Let $\phi(k)$ be the minimum number of vertices in a non-$k$-choosable $k$-chromatic graph. The Ohba conjecture, confirmed by Noel, Reed and Wu, asserts that $\phi(k) \ge 2k+2$. This bound is tight if $k$ is even. If $k$ is odd, then it is…

Given two graphs $G$ and $H$, the $k$-colored Gallai-Ramsey number $gr_k(G : H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of the complete graph on $n$ vertices contains either a rainbow copy of $G$ or a…

Let $G=(V,E)$ be a graph of order $n$ with chromatic number $\chi(G)$. Let $ k \geq \chi(G) $ and $S \subseteq V$. Let $ C_0 $ be a $k$-coloring of the induced subgraph $ G[S] $. The coloring $C_0$ is called an extendable coloring, if $C_0$…

This paper describes how many known graph polynomials arise from the coefficients of chromatic symmetric function expansions in different bases, and studies a new polynomial arising by expanding over a basis given by chromatic symmetric…

Sudoku grids can be thought of as graphs where the vertices are the squares of the grid, and edges join vertices in the same row, column, or sub-grid. A Sudoku puzzle corresponds to a partial proper coloring of the Sudoku graph. We provide…

A decomposition of a non-empty simple graph $G$ is a pair $[G,P]$, such that $P$ is a set of non-empty induced subgraphs of $G$, and every edge of $G$ belongs to exactly one subgraph in $P$. The chromatic index $\chi'([G,P])$ of a…