Related papers: Hamiltonicity of Bell and Stirling Colour Graphs

A harmonious coloring of $G$ is a proper vertex coloring of $G$ such that every pair of colors appears on at most one pair of adjacent vertices. The harmonious chromatic number of $G$, $h(G)$, is the minimum number of colors needed for a…

A graph $G$ is {\em $k$-choosable} if for every assignment of a set $S(v)$ of $k$ colors to every vertex $v$ of $G$, there is a proper coloring of $G$ that assigns to each vertex $v$ a color from $S(v)$. We consider the complexity of…

A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph G and a vertex…

Let $H$ be a fixed graph whose edges are colored red and blue and let $\beta \in [0,1]$. Let $I(H, \beta)$ be the (asymptotically normalized) maximum number of copies of $H$ in a large red/blue edge-colored complete graph $G$, where the…

The spectrum of the $k$-power hypergraph of a graph $G$ is called the $k$-ordered spectrum of $G$.If graphs $G_1$ and $G_2$ have same $k$-ordered spectrum for all positive integer $k\geq2$, $G_1$ and $G_2$ are said to be high-ordered…

A $k$-star colouring of a graph $G$ is a function $f:V(G)\to\{0,1,\dots,k-1\}$ such that $f(u)\neq f(v)$ for every edge $uv$ of $G$, and every bicoloured connected subgraph of $G$ is a star. The star chromatic number of $G$, $\chi_s(G)$, is…

A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. We prove that if cycles of length at most four in a planar graph G are pairwise far apart, then G is 3-choosable. This is analogous…

A $k$-uniform tight cycle is a $k$-graph with a cyclic order of its vertices such that every $k$ consecutive vertices from an edge. We show that for $k\geq 3$, every red-blue edge-coloured complete $k$-graph on $n$ vertices contains $k$…

We investigate the extent to which the $k$-coloring graph $\mathcal{C}_{k}(G)$ uniquely determines the base graph $G$ and the number of colors $k$. The vertices of $\mathcal{C}_{k}(G)$ are the proper $k$-colorings of $G$, and edges connect…

It is well known that the Bell numbers represent the total number of partitions of an n-set. Similarly, the Stirling numbers of the second kind, represent the number of k-partitions of an n-set. In this paper we introduce a certain…

We show that any planar graph $G=(V,E)$ has a 5-coloring such that one color class contains at most $|V|/6$ vertices. In other words, there exists a partition of $V$ into five independent sets $\{V_1, \cdots, V_5\}$ such that $|V_5| \leq…

A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of non-negative integers (with minimum assignment $0$) to its vertices such that the adjacent vertices must receive integers at least two apart and vertices at distance…

We study a new variant of graph coloring by adding a connectivity constraint. A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph $G$ is said to be…

For a graph G, we define a small automorphism as one that maps some vertex into its neighbour. We investigate the edge colourings of G that break every small automorphism of G. We show that such a colouring can be chosen from any set of…

For positive integers n and r we define the Haggkvist-Hell graph, H_{n:r}, to be the graph whose vertices are the ordered pairs (h,T) where T is an r-subset of [n], and h is an element of [n] not in T. Vertices (h_x,T_x) and (h_y,T_y) are…

Given a graph $G$, denote by $\Delta$ and $\chi^\prime$ the maximum degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$\Delta$-critical} if $\chi^\prime(G)=\Delta+1$ and $\chi^\prime(H)\le\Delta$…

Let $F$ be a (possibly improper) edge-coloring of a graph $G$; a vertex coloring of $G$ is \emph{adapted to} $F$ if no color appears at the same time on an edge and on its two endpoints. If for some integer $k$, a graph $G$ is such that…

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…

We consider cell colorings of drawings of graphs in the plane. Given a multi-graph $G$ together with a drawing $\Gamma(G)$ in the plane with only finitely many crossings, we define a cell $k$-coloring of $\Gamma(G)$ to be a coloring of the…

A proper vertex-colouring of a graph G is said to be locally identifying if for any pair u,v of adjacent vertices with distinct closed neighbourhoods, the sets of colours in the closed neighbourhoods of u and v are different. We show that…