Anti‐Ramsey Problems for t Edge‐Disjoint Rainbow Spanning Subgraphs: Cycles, Matchings, or Trees

Abstract

We seek the maximum number of colors in an edge-coloring of the complete graph K n not having t edge-disjoint rainbow spanning subgraphs of specified types. Let c(n, t), m(n, t), and r(n, t) denote the answers when the spanning subgraphs are cycles, matchings, or trees, respectively. We prove c(n, t) = n−1 2 + t for n ≥ 8t − 1 and m(n, t) = n−2 2 + t for n > 4t + 10. We prove r(n, t) = n−2 2 + t for n > 2t + 6t − and r(n, t) = n 2 − t for n = 2t. We also provide constructions for the more general problem in which colorings are restricted so that colors do not appear on more than q edges at a vertex.

Anti-Ramsey Number of Edge-Disjoint Rainbow Spanning Trees

“…Very recently, Jia, Lu, and Zhang [15] considered the analogous problem when the host graph is the complete bipartite graphs K p,q . Using similar approaches in [14], they proved that r(K p,q , 1) = (p − 2)q + 1 + δ pq for p ≥ q ≥ 4 and r(K p,q , t)…”

“…Both of the results in [14] and [15] leave a gap of Θ( √ t) when the number of edge-disjoint rainbow spanning trees is closer to the maximum possible number of edge-disjoint spanning trees in K n and K p,q respectively.…”

“…Akbari and Alipour [1] showed that r(K n , 2) = n−2 2 + 2 for n ≥ 6. Jahanbekam and West [14] extended the investigations to arbitrary number of edge-disjoint rainbow spanning trees (along with the anti-Ramsey number of some other edge-disjoint rainbow spanning structures such as matchings and cycles). In particular, for t edge-disjoint rainbow spanning trees, they showed that r(K n , t)…”

“…and they [14] conjectured that r(K n , t) = n−2 2 + t whenever n ≥ 2t + 2 ≥ 6. This conjecture was recently settled by the first and the third author in [19].…”

Rainbow Spanning Subgraphs of Small Diameter in Edge-Colored Complete Graphs

“…Early results considered the problem with F being a single graph: see [1] for a survey and [4] for the notable determination of AR(n, C k ). More recent work considers problems where F consists of spanning subgraphs of K n ; see [2] for a discussion of such problems involving spanning cycles, perfect matchings, and spanning trees.…”