On Neutral Edge Sets in Anti-Ramsey Numbers

Ali Ghalavand; Qing Jie; Zemin Jin; Xueliang Li; Linshu Pan

Paper

On Neutral Edge Sets in Anti-Ramsey Numbers

Abstract

The anti-Ramsey number of a graph

, introduced by Erdős et al.\ in 1975, is the maximum number of colors in an edge-coloring of the complete graph

that avoids a rainbow copy of

. We call a subset of edges of

\emph{neutral} for the anti-Ramsey number if removing them does not alter the anti-Ramsey number of

. Let

, and

be positive integers, and consider

. Assume

consists of internal edges of the

components in

. It is known that

is neutral when

and

. In this paper, we identify values of

such that, for all

in a specific subinterval of

remains neutral. Since the anti-Ramsey numbers for matchings are well understood, our results provide a complete determination of the anti-Ramsey number for

under these conditions. Based on our findings, we conjecture that this neutrality may extend to the general case

, and

, but not when

, and