Edge-colorings of $K_{m,n}$ which Forbid Multicolored Cycles

TL;DR

This paper investigates proper edge-colorings of complete bipartite graphs that exclude multicolored cycles, establishing conditions under which such cycles must exist and characterizing graphs that forbid specific multicolored cycles.

Contribution

It proves new bounds for the existence of multicolored cycles in properly edge-colored bipartite graphs and characterizes graphs avoiding certain multicolored cycles.

Findings

For any $k extgreater 1$, if $n extgreater 5k-6$, then $K_{k,n}$ contains a multicolored $C_{2k}$.

Determines the order of bipartite graphs that forbid multicolored $C_6$.

Provides conditions linking graph size and cycle multicoloring properties.

Abstract

A subgraph in an edge-colored graph is multicolored if all its edges receive distinct colors. In this paper, we study the proper edge-colorings of the complete bipartite graph $K_{m,n}$ which forbid multicolored cycles. Mainly, we prove that (1) for any integer $k\geq 2$, if $n\geq 5k-6$, then any properly $n$-edge-colored $K_{k,n}$ contains a multicolored $C_{2k}$, and (2) determine the order of the properly edge-colored complete bipartite graphs which forbid multicolored $C_6$.