Related papers: A Lower Bound for Partial List Colorings

For any graph $G$, the First-Fit (or Grundy) chromatic number of $G$, denoted by $\chi_{_{\sf FF}}(G)$, is defined as the maximum number of colors used by the First-Fit (greedy) coloring of the vertices of $G$. We call a family…

The Minimum Sum Coloring Problem is a variant of the Graph Vertex Coloring Problem, for which each color has a weight. This paper presents a new way to find a lower bound of this problem, based on a relaxation into an integer partition…

The packing chromatic number $\chi$ $\rho$ (G) of a graph G is the smallest integer k such that its set of vertices V (G) can be partitioned into k disjoint subsets V 1 ,. .. , V k , in such a way that every two distinct vertices in V i are…

Let $H$ be a graph with $\Delta(H) \leq 2$, and let $G$ be obtained from $H$ by gluing in vertex-disjoint copies of $K_4$. We prove that if $H$ contains at most one odd cycle of length exceeding $3$, or if $H$ contains at most $3$…

A vertex coloring of a graph is said to be \textit{conflict-free} with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al.,…

The Petersen colouring conjecture states that every bridgeless cubic graph admits an edge-colouring with $5$ colours such that for every edge $e$, the set of colours assigned to the edges adjacent to $e$ has cardinality either $2$ or $4$,…

The quantum chromatic number, $\chi_q(G)$, of a graph $G$ was originally defined as the minimal number of colors necessary in a quantum protocol in which two provers that cannot communicate with each other but share an entangled state can…

A proper edge colouring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colours. Using a clever application of the Local Lemma, Hatami (2005) proved that every graph with maximum degree $\Delta$…

A graph \( G \) is said to be (vertex) non-repetitively colored if no simple path in \( G \) has a sequence of vertex colors that forms a repetition. Formally, a coloring \( c: V(G) \to \{1, 2, \dots, k\} \) is non-repetitive if, for every…

We classify the trees on $n$ vertices with the maximum and the minimum number of certain generalized colorings, including conflict-free, odd, non-monochromatic, star, and star rainbow vertex colorings. We also extend a result of Cutler and…

A graph $G$ is equitably $k$-list arborable if for any $k$-uniform list assignment $L$, there is an equitable $L$-colouring of $G$ whose each colour class induces an acyclic graph. The smallest number $k$ admitting such a coloring is named…

A strong edge-coloring of a graph $G$ is an edge-coloring such that no two edges of distance at most two receive the same color. The strong chromatic index $\chi'_s(G)$ is the minimum number of colors in a strong edge-coloring of $G$. P.…

We present a novel algorithm for edge-coloring of multigraphs. The correctness of this algorithm for multigraphs with $\chi' > \Delta +1$ ($\chi'$ is the chromatic edge number and $\Delta$ is the maximum vertex degree) would prove a long…

The \textit{square} of a graph $G$, denoted by $G^2$, is obtained from $G$ by adding an edge to connect every pair of vertices with a common neighbor in $G$. In this paper we prove that for every planar graph $G$ with maximum degree at most…

A graph $G$ is class II, if its chromatic index is at least $\Delta+1$. Let $H$ be a maximum $\Delta$-edge-colorable subgraph of $G$. The paper proves best possible lower bounds for $\frac{|E(H)|}{|E(G)|}$, and structural properties of…

Let $G$ be a graph whose each component has order at least 3. Let $s : E(G) \rightarrow \mathbb{Z}_k$ for some integer $k\geq 2$ be an improper edge coloring of $G$ (where adjacent edges may be assigned the same color). If the induced…

It's well known that every planar graph is $4$-colorable. A toroidal graph is a graph that can be embedded on a torus. It's proved that every toroidal graph is $7$-colorable. A proper coloring of a graph is called \emph{odd} if every…

The strong chromatic number $\chi_{\text{s}}(G)$ of a graph $G$ on $n$ vertices is the least number $r$ with the following property: after adding $r \lceil n/r \rceil - n$ isolated vertices to $G$ and taking the union with any collection of…

We show a method how to convert any graph into the binary number and vice versa. We derive upper bound for maximum number of graphs, that, have fixed number of vertices and can be colored with n colors (n is any given number). Proof for the…

A colored complete graph is said to be Gallai-colored if it contains no rainbow triangle. This property has been shown to be equivalent to the existence of a partition of the vertices (of every induced subgraph) in which at most two colors…