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Journal Articles Discrete Applied Mathematics Year : 2022

Smallest $C_{2l+1}$-critical graphs of odd-girth $2k+1$

Abstract

Given a graph H, a graph G is called H-critical if G does not admit a homomorphism to H, but any proper subgraph of G does. Observe that K k−1-critical graphs are the standard k-(colour)-critical graphs. We consider questions of extremal nature previously studied for k-critical graphs and generalize them to H-critical graphs. After complete graphs, the next natural case to consider for H is that of the odd-cycles. Thus, given integers and k, ≥ k, we ask: what is the smallest order of a C 2 +1-critical graph of odd-girth at least 2k + 1? Denoting this value by η(k, C 2 +1), we show that η(k, C 2 +1) = 4k for 1 ≤ ≤ k ≤ 3 +i−3 2 (2k = i mod 3) and that η(3, C 5) = 15. The latter means that a smallest graph of odd-girth 7 not admitting a homomorphism to the 5-cycle is of order 15. Computational work shows that there are exactly eleven such graphs on 15 vertices.
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Dates and versions

hal-03784719 , version 1 (23-09-2022)

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Laurent Beaudou, Florent Foucaud, Reza Naserasr. Smallest $C_{2l+1}$-critical graphs of odd-girth $2k+1$. Discrete Applied Mathematics, 2022, 319, pp.564 - 575. ⟨10.1016/j.dam.2021.08.040⟩. ⟨hal-03784719⟩
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