Core Concepts
The authors asymptotically determine the proper rainbow saturation number of the 4-cycle, and provide improved upper bounds for the proper rainbow saturation numbers of the 5-cycle and 6-cycle.
Abstract
The content discusses the study of proper rainbow saturation numbers, which is a recently introduced concept in extremal graph theory. Given a fixed graph F, the proper rainbow saturation number sat*(n, F) is the minimum number of edges in an n-vertex graph that is rainbow F-saturated, meaning it admits a proper edge-coloring that avoids any rainbow copy of F, but the addition of any edge would create such a rainbow copy.
The key results are:
The authors asymptotically determine sat*(n, C4), showing that it is bounded above by 11/6 n + O(1) and below by (11/6 - ε)n for any ε > 0 and sufficiently large n. This separates sat*(n, C4) from the ordinary saturation number sat(n, C4) by a constant multiplicative factor.
The authors provide improved upper bounds on sat*(n, C5) and sat*(n, C6):
sat*(n, C5) ≤ ⌊5n/2⌋ - 4 for n ≥ 9
sat*(n, C6) ≤ 7/3 n + O(1)
The authors establish structural properties of rainbow C4-saturated graphs, showing that any such graph either has a small dominating set or a dominating set with high average degree. This structural insight is crucial for their lower bound on sat*(n, C4).
The authors also discuss general results on the behavior of proper rainbow saturation numbers, including a theorem that for graphs F containing no induced even cycles, sat*(n, F) is linear in n.
Stats
n ≥ 7
sat*(n, C4) ≤ 11/6 n + O(1)
n ≥ 9
sat*(n, C5) ≤ ⌊5n/2⌋ - 4
sat*(n, C6) ≤ 7/3 n + O(1)
Quotes
"We say that an edge-coloring of a graph G is proper if every pair of incident edges receive distinct colors, and is rainbow if no two edges of G receive the same color."
"Given fixed graphs G, F, and a proper edge-coloring c of G, we say that G is rainbow-F-free under c if G does not contain any copy of F which is rainbow with respect to c."
"We call sat*(n, F) the proper rainbow saturation number of F, since all edge-colorings in this setting are proper."