Some extremal results on the chromatic-stability index

Abstract

The χ-stability index es χ (G) of a graph G is the minimum number of its edges whose removal results in a graph with the chromatic number smaller than that of G. In this paper three open problems from [European J. Combin. 84 (2020) 103042] are considered. Examples are constructed which demonstrate that a known characterization of k-regular (k ≤ 5) graphs G with es χ (G) = 1 does not extend to k ≥ 6. Graphs G with χ(G) = 3 for which es χ (G) + es χ (G) = 2 holds are characterized. Necessary conditions on graphs G which attain a known upper bound on es χ (G) in terms of the order and the chromatic number of G are derived. The conditions are proved to be sufficient when n ≡ 2 (mod 3) and χ(G) = 3.

On the Chromatic Vertex Stability Number of Graphs

“…The paper [1] brings Nordhaus-Gaddum type inequality for es χ (stronger than a related result from [4]), sharp upper bounds on es χ in terms of size and of maximum degree, and a characterization of graphs with es χ = 1 among k-regular graphs for k ≤ 5. 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].…”