Related papers: Odd graph and its applications to the strong edge …

A coloring of the edges of a graph $G$ in which every $K_{1,2}$ is totally multicolored is known as a proper coloring and a coloring of the edges of $G$ in which every $K_{1,2}$ and every $K_{2,2}$ is totally multicolored is called a…

A total coloring of a graph $G$ is a coloring of the vertices and edges such that two adjacent or incident elements receive different colors. The minimum number of colors required for a total coloring of a graph $G$ is called the total…

In this paper we show that every graph $G$ of bounded maximum average degree ${\rm mad}(G)$ and with maximum degree $\Delta$ can be edge-colored using the optimal number of $\Delta$ colors in quasilinear time, whenever $\Delta\ge 2{\rm…

Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a…

Let $G$ be a graph with $n$ vertices, $m$ edges, average degree $\delta$, and maximum degree $\Delta$. The "oriented chromatic number" of $G$ is the maximum, taken over all orientations of $G$, of the minimum number of colours in a proper…

We say a graph is $(d, d, \ldots, d, 0, \ldots, 0)$-colorable with $a$ of $d$'s and $b$ of $0$'s if $V(G)$ may be partitioned into $b$ independent sets $O_1,O_2,\ldots,O_b$ and $a$ sets $D_1, D_2,\ldots, D_a$ whose induced graphs have…

Let $G$ be a graph and $k$ a positive integer. A strong $k$-edge-coloring of $G$ is a mapping $\phi: E(G)\to \{1,2,\dots,k\}$ such that for any two edges $e$ and $e'$ that are either adjacent to each other or adjacent to a common edge,…

A proper vertex $k$-coloring of a graph $G=(V,E)$ is an assignment $c:V\to \{1,2,\ldots,k\}$ of colors to the vertices of the graph such that no two adjacent vertices are associated with the same color. The square $G^2$ of a graph $G$ is…

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…

An edge-coloring of a graph $G$ assigns a color to each edge of $G$. An edge-coloring is a parity edge-coloring if for each path $P$ in $G$, it uses some color on an odd number of edges in $P$. It is a strong parity edge-coloring if for…

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that for any vertex $v$, there are at least $\min\{r,\deg_G(v) \}$ distinct colors in $N_G(v)$. The $r$-dynamic chromatic number $\chi_r^d(G)$ of a graph $G$ is the…

A signed graph is a simple graph with two types of edges: positive and negative edges. Switching a vertex $v$ of a signed graph corresponds to changing the type of each edge incident to $v$. A homomorphism from a signed graph $G$ to another…

Given a graph $G$, an injective edge-coloring of $G$ is a function $\psi:E(G) \rightarrow \mathbb N$ such that if $\psi(e) = \psi(e')$, then no third edge joins an endpoint of $e$ and an endpoint of $e'$. The injective chromatic index of a…

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors in an acyclic edge…

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…

Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors $k$ that can ensure an odd coloring of a…

An edge coloring of a graph $G$ is to color all the edges in the graph such that adjacent edges receive different colors. It is acyclic if each cycle in the graph receives at least three colors. Fiam{\v{c}}ik (1978) and Alon, Sudakov and…

We study the linear list chromatic number, denoted $\lcl(G)$, of sparse graphs. The maximum average degree of a graph $G$, denoted $\mad(G)$, is the maximum of the average degrees of all subgraphs of $G$. It is clear that any graph $G$ with…

We prove that any class of graphs with linear neighborhood complexity has bounded improper odd chromatic number. As a result, if $\mathcal{G}$ is the class of all circle graphs, or if $\mathcal{G}$ is any class with bounded twin-width,…

We show that for any fixed integer $m \geq 1$, a graph of maximum degree $\Delta$ has a coloring with $O(\Delta^{(m+1)/m})$ colors in which every connected bicolored subgraph contains at most $m$ edges. This result unifies previously known…