Related papers: Odd Properly Colored Cycles in Edge-Colored Graphs

A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. A full house is a graph composed by a vertex adjacent to both ends of an edge in $K_4$ . Let $H$ be the complement of a cycle on 7 vertices.…

In this paper, we investigate the ratio of the numbers of odd and even cycles in outerplanar graphs. We verify that the ratio generally diverges to infinity as the order of a graph diverges to infinity. We also give sharp estimations of the…

A proper coloring of a graph $G$ is said to be a strong odd coloring of $G$, if for every vertex $v$ and every color $c$, either $c$ appears on an odd number of vertices in the neighborhood of $v$ or $c$ is absent in the neighborhood of…

In this note, we prove an interesting result about perfect matchings in a complete bipartite graph with 2n vertices on each side, whose edges are colored in red and blue such that each vertex is part of n red edges and n blue edges.

Let $k$ and $r$ be two integers with $k \ge 2$ and $k\ge r \ge 1$. In this paper we show that (1) if a strongly connected digraph $D$ contains no directed cycle of length $1$ modulo $k$, then $D$ is $k$-colorable; and (2) if a digraph $D$…

We prove that graphs that do not contain a totally odd immersion of $K_t$ are $\mathcal{O}(t)$-colorable. In particular, we show that any graph with no totally odd immersion of $K_t$ is the union of a bipartite graph and a graph which…

A proper edge coloring of a graph without any bichromatic cycles is said to be an acyclic edge coloring of the graph. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum integer $k$ such that $G$ has an acyclic…

We say that a vertex or edge colouring of a graph is distinguishing if the only automorphism that preserves this colouring is the identity. A (proper) distinguishing colouring is irreducible if there is no possibility of merging two…

We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption…

An edge-weighting of a graph is called vertex-coloring if the weighted degrees yield a proper vertex coloring of the graph. It is conjectured that for every graph without isolated edge, a vertex-coloring edge-weighting with the set {1,2,3}…

A conjecture due to the fourth author states that every $d$-regular planar multigraph can be $d$-edge-coloured, provided that for every odd set $X$ of vertices, there are at least $d$ edges between $X$ and its complement. For $d = 3$ this…

A proper vertex coloring $\varphi$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $\varphi^{-1}(c)\cap N(x)$ is odd-sized. The minimum number of colors in any odd coloring of…

Let $G$ be an edge-colored graph. A heterochromatic cycle of $G$ is one in which every two edges have different colors. For a vertex $v\in V(G)$, let $CN(v)$ denote the set of colors which are assigned to the edges incident to $v$. In this…

We define a method for edge coloring signed graphs and what it means for such a coloring to be proper. Our method has many desirable properties: it specializes to the usual notion of edge coloring when the signed graph is all-negative, it…

It was conjectured by the third author in about 1973 that every $d$-regular planar graph (possibly with parallel edges) can be $d$-edge-coloured, provided that for every odd set $X$ of vertices, there are at least $d$ edges between $X$ and…

We consider edge colorings of graphs. An edge coloring is a majority coloring if for every vertex at most half of the edges incident with it are in one color. And edge coloring is a distinguishing coloring if for every non-trivial…

A strong edge coloring of a graph $G$ is a proper edge coloring in which each color class is an induced matching of $G$. In 1993, Brualdi and Quinn Massey proposed a conjecture that every bipartite graph without $4$-cycles and with the…

It is proved that there exists an absolute constant c > 0 such that for every natural number k, every non-bipartite 2-connected graph with average degree at least ck contains k cycles with consecutive odd lengths. This implies the existence…

A bipartite graph is called bipancyclic if it contains cycles of every even length from four up to the number of vertices in the graph. A theorem of Schmeichel and Mitchem states that for $n \geq 4$, every balanced bipartite graph on $2n$…

We prove that a complete bipartite graph can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, the length of each cycle is at most the size of the smallest part, and the…