On the chromatic edge stability index of graphs

Abstract

Given a non-trivial graph G, the minimum cardinality of a set of edges F in G such that χ ′ (G \ F ) < χ ′ (G) is called the chromatic edge stability index of G, denoted by es χ ′ (G), and such a (smallest) set F is called a (minimum) mitigating set. While 1 ≤ es χ ′ (G) ≤ ⌊n/2⌋ holds for any graph G, we investigate the graphs with extremal and near-extremal values of es χ ′ (G). The graphs G with es χ ′ (G) = ⌊n/2⌋ are classified, and the graphs G with es χ ′ (G) = ⌊n/2⌋ − 1 and χ ′ (G) = ∆(G) + 1 are characterized. We establish that the odd cycles and K2 are exactly the regular connected graphs with the chromatic edge stability index 1; on the other hand, we prove that it is NP-hard to verify whether a graph G has es χ ′ (G) = 1. We also prove that every minimum mitigating set of an r-regular graph G, where r = 4, with es χ ′ (G) = 2 is a matching. Furthermore, we propose a conjecture that for every graph G there exists a minimum mitigating set, which is a matching, and prove that the conjecture holds for graphs G with es χ ′ (G) ∈ {1, 2, ⌊n/2⌋ − 1, ⌊n/2⌋}, and for bipartite graphs.

On the Chromatic Vertex Stability Number of Graphs

“…In [6] progress on three open problems from [1] are reported. The chromatic edge stability number has been generalized to arbitrary graphical invariants in [7], where in particular it was considered with respect to the chromatic index, see also [2,3].…”