On the strong chromatic index and induced matching of tree-cographs, permutation graphs and chordal bipartite graphs (original) (raw)

Strong chromatic index of graphs: a short survey

A strong edge coloring of a graph G is an edge coloring such that every two adjacent edges or two edges adjacent to a same edge receive two distinct colors; in other words, every path of length three has three distinct colors in G. The strong chromatic index of G, denoted by , is the smallest integer k such that G admits a strong edge coloring with k colors. This survey is an brief introduction to some good results regarding the strong chromatic index of planar graphs, bipartite graphs and so on.

Minimum Cost Edge-Colorings of Trees Can Be Reduced to Matchings

IEICE Transactions on Information and Systems, 2011

Let C be a set of colors, and let ω(c) be an integer cost assigned to a color c in C. An edge-coloring of a graph G is to color all the edges of G so that any two adjacent edges are colored with different colors in C. The cost ω(f) of an edge-coloring f of G is the sum of costs ω(f (e)) of colors f (e) assigned to all edges e in G. An edge-coloring f of G is optimal if ω(f) is minimum among all edge-colorings of G. In this paper, we show that the problem of finding an optimal edge-coloring of a tree T can be simply reduced in polynomial time to the minimum weight perfect matching problem for a new bipartite graph constructed from T. The reduction immediately yields an efficient simple algorithm to find an optimal edge-coloring of T in time O(n 1.5 Δ log(nN ω)), where n is the number of vertices in T , Δ is the maximum degree of T , and N ω is the maximum absolute cost |ω(c)| of colors c in C. We then show that our result can be extended for multitrees.

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

The Electronic Journal of Combinatorics, 2021

A star edge coloring of a graph GGG is a proper edge coloring of GGG such that every path and cycle of length four in GGG uses at least three different colors. The star chromatic index of a graph GGG is the smallest integer kkk for which GGG admits a star edge coloring with kkk 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.

On the computational complexity of matching on chordal and strongly chordal graphs

1994

In this paper we study the computational complexity (both sequential and parallel) of the maximum matching problem for chordal and strongly chordal graphs. We show that there is a linear time greedy algorithm for a maximum matching in a strongly chordal graph provided a strongly perfect elimination ordering is known. This algorithm can be also turned into a parallel algorithm. The technique used can be also extended for the multidimensional matching for chordal and strongly chordal graphs yielding the rst polynomial time algorithms for these classes of graphs (the multidimensional matching is NPcomplete in general).

The Parallel and Sequential Complexity of Matching on Chordal and Strongly Chordal Graphs

1995

Chordal graphs became interesting as a generalization of interval graphs (see for example \cite{LB}). We call a graph chordal if every cycle of length greater than three has a chord, i.e. an edge that joins two non consecutive vertices of the cycle. Note that interval graphs are not only chordal but strongly chordal as defined in \cite{Fa1}. Strongly chordal graphs are just those chordal graphs having a so called strongly perfect elimination ordering. In this paper we consider the sequential and parallel complexity of the maximum matching problem in chordal and strongly chordal graphs. Note that in general a linear time algorithm for perfect matching is not known. Here we shall show that, provided a strongly perfect elimination ordering is known, a maximum matching in a strongly chordal graph can be found in linear time by a simple greedy algorithm. This algorithm can be turned into a (non optimal) parallel algorithm.

On the Sequential and Parallel Complexity of Matching in Chordal and Strongly Chordal Graphs

In this paper we consider the sequential and parallel complexity of the maximum matching problem in chordal and strongly chordal graphs. We prove that, given a strongly perfect elimination ordering, a maximum matching in a strongly chordal graph can be found in a linear time. On the other hand we observe that the matching problem restricted to chordal (paths) graphs is of the same parallel complexiy as a general bipartite matching.

A linear-time algorithm for the strong chromatic index of Halin graphs

We show that there exists a linear-time algorithm that computes the strong chromatic index of Halin graphs. In their paper Molloy and Reed state that ǫ 0.002 when ∆ is sufficiently large. 4 This algorithm checks in O(n(s + 1) t ) time whether a partial k-tree has a strong edge coloring that uses at most s colors. Here, the exponent t = 2 4(k+1)+1 .

Strong chromatic index of products of graphs

Discrete mathematics & theoretical computer science DMTCS

The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for strong chromatic index of three different products of graphs in term of the strong chromatic index of each factor. For the cartesian product of paths, cycles or complete graphs, we derive sharper results. In particular, strong chromatic indices of d-dimensional grids and of some toroidal grids are given along with approximate results on the strong chromatic index of generalized hypercubes.