2 research outputs found
Existences of rainbow matchings and rainbow matching covers
Let be an edge-coloured graph. A rainbow subgraph in is a subgraph
such that its edges have distinct colours. The minimum colour degree
of is the smallest number of distinct colours on the edges
incident with a vertex of . We show that every edge-coloured graph on
vertices with contains a rainbow matching
of size at least , which improves the previous result for .
Let be the maximum number of edges of the same
colour incident with a vertex of . We also prove that if and
, then can be edge-decomposed into at most
rainbow matchings. This result is sharp and improves a
result of LeSaulnier and West
Rainbow Perfect and Near-Perfect Matchings in Complete Graphs with Edges Colored by Circular Distance
Given an edge-colored complete graph Kn on n vertices, a perfect (respectively, near-perfect) matching M in Kn with an even (respectively, odd) number of vertices is rainbow if all edges have distinct colors. In this paper, we consider an edge coloring of Kn by circular distance, and we denote the resulting complete graph by K●n. We show that when K●n has an even number of vertices, it contains a rainbow perfect matching if and only if n=8k or n=8k+2, where k is a nonnegative integer. In the case of an odd number of vertices, Kirkman matching is known to be a rainbow near-perfect matching in K●n. However, real-world applications sometimes require multiple rainbow near-perfect matchings. We propose a method for using a recursive algorithm to generate multiple rainbow near-perfect matchings in K●n