Strong chromatic index of generalized Jahangir graphs and generalized Helm graphs

The strong edge coloring of a graph G is a proper edge coloring that assigns a different color to any two edges which are at most two edges apart. The minimum number of color classes that contribute to such a proper coloring is said to be the strong chromatic index of G. This paper defines the strong chromatic index for the generalized Jahangir graphs and the generalized Helm graphs.

The Dominator Edge Coloring of Graphs

Let G be a simple graph. A dominator edge coloring (DE-coloring) of G is a proper edge coloring in which each edge of G is adjacent to every edge of some color class (possibly its own class). The dominator edge chromatic number (DEC-number) of G is the minimum number of color classes among all dominator edge colorings of G , denoted by χ d ′ G . In this paper, we establish the bounds of the DEC-number of a graph, present the DEC-number of special graphs, and study the relationship of the DEC-number between G and the operations of G .

TWO RESULTS ON THE PALETTE INDEX OF GRAPHS

Given a proper edge coloring $\alpha$ of a graph $G$, we define the palette $S_G(v,\alpha)$ of a vertex $v\in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check{s}(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. A graph $G$ is called nearly bipartite if there exists $ v\in V(G)$ so that $G-v$ is a bipartite graph. In this paper, we give an upper bound on the palette index of a nearly bipartite graph $G$ by using the decomposition of $G$ into cycles. We also provide an upper bound on the palette index of Cartesian products of graphs. In particular, we show that for any graphs $G$ and $H$, $\check{s}(G\square H)\leq \check{s}(G)\check{s}(H)$.

Restricted Extension of Sparse Partial Edge Colorings of Complete graphs

Given a partial edge coloring of a complete graph $K_n$ and lists of allowed colors for the non-colored edges of $K_n$, can we extend the partial edge coloring to a proper edge coloring of $K_n$ using only colors from the lists? We prove that this question has a positive answer in the case when both the partial edge coloring and the color lists satisfy certain sparsity conditions.

A Polynomial Time Algorithm to Find the Star Chromatic Index of Trees

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of a graph $G$ is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors. In this paper, we present a polynomial time algorithm that finds an optimum star edge coloring for every tree. We also provide some tight bounds on the star chromatic index of trees with diameter at most four, and using these bounds we find a formula for the star chromatic index of certain families of trees.

Introduction to dominated edge chromatic number of a graph

We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph \(G\), is a proper edge coloring of \(G\) such that each color class is dominated by at least one edge of \(G\). The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by \(\chi_{dom}^{\prime}(G)\). We obtain some properties of \(\chi_{dom}^{\prime}(G)\) and compute it for specific graphs. Also examine the effects on \(\chi_{dom}^{\prime}(G)\), when \(G\) is modified by operations on vertex and edge of \(G\). Finally, we consider the \(k\)-subdivision of \(G\) and study the dominated edge chromatic number of these kind of graphs.

Strong Edge Coloring of Generalized Petersen Graphs

A strong edge coloring of a graph G is a proper edge coloring such that every color class is an induced matching. In 2018, Yang and Wu proposed a conjecture that every generalized Petersen graph P(n,k) with k≥4 and n>2k can be strong edge colored with (at most) seven colors. Although the generalized Petersen graph P(n,k) is a kind of special graph, the strong chromatic index of P(n,k) is still unknown. In this paper, we support the conjecture by showing that the strong chromatic index of every generalized Petersen graph P(n,k) with k≥4 and n>2k is at most 9.

Adjacent vertex distinguishing edge coloring of planar graphs without 3-cycles

The adjacent vertex distinguishing edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that the color sets of any pair of adjacent vertices are distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of [Formula: see text] is denoted by [Formula: see text]. It is observed that [Formula: see text] when [Formula: see text] contains two adjacent vertices of degree [Formula: see text]. In this paper, we prove that if [Formula: see text] is a planar graph without 3-cycles, then [Formula: see text]. Furthermore, we characterize the adjacent vertex distinguishing chromatic index for planar graphs of [Formula: see text] and without 3-cycles. This improves a result from [D. Huang, Z. Miao and W. Wang, Adjacent vertex distinguishing indices of planar graphs without 3-cycles, Discrete Math. 338 (2015) 139–148] that established [Formula: see text] for planar graphs without 3-cycles.

The Rainbow Connection Number of Triangular Snake Graphs

Rainbow coloring problems, of noteworthy applications in Information Security, have been receiving much attention last years in Combinatorics. The rainbow connection number of a graph G is the least number of colors for a (not necessarily proper) edge coloring of G such that between any pair of vertices there is a path whose edge colors are all distinct. In this paper we determine the rainbow connection number of the triple triangular snake graphs.