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SOME EXTREMAL RESULTS ON THE CHROMATIC STABILITY INDEX
Part of:
Graph theory
Published online by Cambridge University Press: 04 March 2022
Abstract
The
$\chi $
-stability index
$\mathrm {es}_{\chi }(G)$
of a graph G is the minimum number of its edges whose removal results in a graph with chromatic number smaller than that of G. We consider three open problems from Akbari et al. [‘Nordhaus–Gaddum and other bounds for the chromatic edge-stability number’, European J. Combin. 84 (2020), Article no. 103042]. We show by examples that a known characterisation of k-regular (
$k\le 5$
) graphs G with
$\mathrm {es}_{\chi }(G) = 1$
does not extend to
$k\ge 6$
, and we characterise graphs G with
$\chi (G)=3$
for which
$\mathrm { es}_{\chi }(G)+\mathrm {es}_{\chi }(\overline {G}) = 2$
. We derive necessary conditions on graphs G which attain a known upper bound on
$\mathrm { es}_{\chi }(G)$
in terms of the order and the chromatic number of G and show that the conditions are sufficient when
$n\equiv 2 \pmod 3$
and
$\chi (G)=3$
.
MSC classification
Secondary:
05C35: Extremal problems
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 106 , Issue 2 , October 2022 , pp. 185 - 194
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
Huang was partially supported by the National Natural Science Foundation of China (11801284) and the Fundamental Research Funds for the Central Universities, Nankai University. Lei, Lian and Shi were partially supported by the China–Slovenia bilateral project ‘Some topics in modern graph theory’ (No. 12-6), the National Natural Science Foundation of China and the Fundamental Research Funds for the Central Universities, Nankai University. Klavžar acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0297 and projects J1-9109, J1-1693, N1-0095, N1-0108).
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