Related papers: A note on graphs of $k$-colourings

Let $K_4^+$ be the 5-vertex graph obtained from $K_4$, the complete graph on four vertices, by subdividing one edge precisely once (i.e. by replacing one edge by a path on three vertices). We prove that if the chromatic number of some graph…

A $t$-tone $k$-coloring of $G$ assigns to each vertex of $G$ a set of $t$ colors from $\{1,..., k\}$ so that vertices at distance $d$ share fewer than $d$ common colors. The {\it $t$-tone chromatic number} of $G$, denoted $\tau_t(G)$, is…

The chromatic polynomial $P(G,x)$ of a graph $G$ of order $n$ can be expressed as $\sum\limits_{i=1}^n(-1)^{n-i}a_{i}x^i$, where $a_i$ is interpreted as the number of broken-cycle free spanning subgraphs of $G$ with exactly $i$ components.…

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices. A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but…

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have…

Let $G$ be a graph of maximum degree $\Delta$ and $k$ be an integer. The $k$-recolouring graph of $G$ is the graph whose vertices are $k$-colourings of $G$ and where two $k$-colourings are adjacent if they differ at exactly one vertex. It…

A proper edge $k$-colouring of a graph $G=(V,E)$ is an assignment $c:E\to \{1,2,\ldots,k\}$ of colours to the edges of the graph such that no two adjacent edges are associated with the same colour. A neighbour sum distinguishing edge…

Let $G$ be a graph and $r\in\mathbb{N}$. The matching Kneser graph $\textsf{KG}(G, rK_2)$ is a graph whose vertex set is the set of $r$-matchings in $G$ and two vertices are adjacent if their corresponding matchings are edge-disjoint. In…

A rainbow path in an edge coloured graph is a path in which no two edges are coloured the same. A rainbow colouring of a connected graph G is a colouring of the edges of G such that every pair of vertices in G is connected by at least one…

A k-role coloring of a graph G is an assignment of k colors to the vertices of G such that if any two vertices are assigned the same color, then their neighborhood are assigned the same set of colors. By definition, every graph on n…

A graph $G$ is said to be $k$-distinguishable if the vertex set can be colored using $k$ colors such that no non-trivial automorphism fixes every color class, and the distinguishing number $D(G)$ is the least integer $k$ for which $G$ is…

In this paper, we introduce the notion of $t$-tone edge coloring. A $t$-tone edge $k$-coloring of a graph $G$ assigns to each edge of $G$ a set of $t$ distinct colors from $\{1,\dots,k\}$ such that any two edges at distance $d$ share fewer…

For a graph $G$ and an integer $k\geq 2$, a $\chi'_{k}$-coloring of $G$ is an edge coloring of $G$ such that the subgraph induced by the edges of each color has all degrees congruent to $1 ~ (\mod k)$, and $\chi'_{k}(G)$ is the minimum…

A $k$-subcolouring of a graph $G$ is a function $f:V(G) \to \{0,\ldots,k-1\}$ such that the set of vertices coloured $i$ induce a disjoint union of cliques. The subchromatic number, $\chi_{\textrm{sub}}(G)$, is the minimum $k$ such that $G$…

We prove that if the edges of a graph G can be colored blue or red in such a way that every vertex belongs to a monochromatic k-clique of each color, then G has at least 4(k-1) vertices. This confirms a conjecture of Bucic, Lidicky, Long,…

The reconfiguration graph for the $k$-colourings of a graph $G$, denoted $R_{k}(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on exactly one vertex. For any…

A Gallai coloring of a complete graph $K_n$ is an edge coloring without triangles colored with three different colors. A sequence $e_1\ge \dots \ge e_k$ of positive integers is an $(n,k)$-sequence if $\sum_{i=1}^k e_i=\binom{n}{2}$. An…

For integers $k, r > 0$, a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to at least $\min\{r, d(v)\}$ differently colored…

Properly colored cycles in edge-colored graphs are closely related to directed cycles in oriented graphs. As an analogy of the well-known Caccetta-H\"{a}ggkvist Conjecture, we study the existence of properly colored cycles of bounded length…

For a positive integer $k$ and graph $G=(V,E)$, a $k$-colouring of $G$ is a mapping $c: V\rightarrow\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The $k$-Colouring problem is to decide, for a given $G$, whether a…