Anti-Ramsey number of matchings in outerplanar graphs

Abstract

A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. Given a graph family G and a graph H, the anti-Ramsey number AR(G,H) is the maximum number of colors in an edge-coloring of a graph G∈G which has no rainbow copy of H. When G={G}, we write AR(G,H) for short. Erdős, Simonovits and Sós introduced these numbers in the 1970s, they studied the case when G is a complete graph. Since then, many results have appeared considering G and H belonging to specific graph classes. In particular, Jendrol’, Schiermeyer and Tu (Jendrol’ et al., 2014) obtained an upper bound for AR(Tn,kK2) in terms of n and k for k≥3 and n≥2k, where Tn is the family of maximal planar graphs on n vertices. Jin and Ye (Jin and Ye, 2018) continued the study of anti-Ramsey numbers of matchings in planar graphs, by deriving an upper bound for AR(Mn,kK2), where Mn is the family of maximal outerplanar graphs on n vertices. In both works, examples are shown for which there are colorings using many colors while avoiding rainbow matchings, but the numbers of colors in these examples do not reach the upper bounds. The bounds of AR(Tn,kK2) were improved in following works by other researchers. This paper goes in the same direction of these two works, proving exact value for AR(Bn,kK2) in terms of n and k for n≥6 and k≥3, where Bn is the family of maximal bipartite outerplanar graphs on n vertices which contains a Hamilton cycle.