Rainbow Perfect and Near-Perfect Matchings in Complete Graphs with Edges Colored by Circular Distance
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Given an edge-colored complete graph K_n on n vertices, a perfect (respectively, near-perfect) matching M in K_n 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 K_n 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.
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