Complete solution for the rainbow numbers of matchings

Abstract

For a given graph H and n ≥ 1, let f (n, H) denote the maximum number c for which there is a way to color the edges of the complete graph K n with c colors such that every subgraph H of K n has at least two edges of the same color. Equivalently, any edge-coloring of K n with at least rb(n, H) = f (n, H) + 1 colors contains a rainbow copy of H, where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number rb(n, H) is called the rainbow number of H. Erdős, Simonovits and Sós showed that rb(n, K 3 ) = n. In 2004, Schiermeyer used some counting technique and determined the rainbow numbers rb(n, kK 2 ) for k ≥ 2 and n ≥ 3k + 3. It is easy to see that n must be at least 2k. So, for 2k ≤ n < 3k + 3, the rainbow numbers remain not determined. In this paper we will use the Gallai-Edmonds structure theorem for matchings to determine the exact values for rainbow numbers rb(n, kK 2 ) for all k ≥ 2 and n ≥ 2k, giving a complete solution for the rainbow numbers of matchings.

Anti-Ramsey Numbers of Paths and Cycles in Hypergraphs

“…is a bad P k−1 in F 0 , a contradiction to (5). but v is an isolated vertex in G * , then we have at least n−(t−1)−1 s−2 > ⌊n/s⌋ + 2(k − 1) − 2 edges meeting L but not belonging to F 0 , which is a contradiction.…”

“…They proved that ex(n, s, M k−1 ) + 2 ≤ ar(n, s, M k ) ≤ ex(n, s, M k−1 ) + k, where the lower bound holds for every n, and the upper bound holds for n ≥ sk + (s − 1)(k − 1). For s = 2, Schiermeyer [40] proved that ar(n, 2, M k ) = ex(n, 2, M k−1 ) + 2 for k ≥ 2 and n ≥ 3k + 3, and this condition was further released to all n ≥ 2k + 1 by Chen, Li and Tu [5] and by Fujita, Kaneko, Schiermeyer and Suzuki [17], independently.…”

Anti‐Ramsey numbers of doubly edge‐critical graphs

“…Anti-Ramsey numbers were introduced by Erdős et al [8]. Various results about this extremal function have been obtained since then: [1,10,14,11,2,12,17,15,13,16,4] to name a few.…”

Improved bounds for rainbow numbers of matchings in plane triangulations

“…. , v q }) = ∅ since otherwise T n contains a K 3,3 -minor (with one part {v 4 , v 5 , v 6 } and the other part 4 , v j } contains two edge-disjoint 3K 2 , say M 1 and M 2 . Assume c(v j v 2 ) = c(e) for some M 1 .…”

“…Hence, v j u i ∈ E(T n ) for some i ∈ [3] and j ≥ 5. 4 , v 5 } contains two edgedisjoint 3K 2 , say M 1 and M 2 . Assume c(v 5 v 2 ) = c(e) for some M 1 .…”