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arXiv · 2301.10190

Chv\'atal-Erd\H{o}s condition for pancyclicity

Nemanja Draganić·David Munhá Correia·Benny Sudakov
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Abstract

An $n$-vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices and it is pancyclic if it contains cycles of all lengths from $3$ up to $n$. A celebrated meta-conjecture of Bondy states that every non-trivial condition implying Hamiltonicity also implies pancyclicity (up to possibly a few exceptional graphs). We show that every graph $G$ with $\kappa(G) > (1+o(1)) \alpha(G)$ is pancyclic. This extends the famous Chv\'atal-Erd\H{o}s condition for Hamiltonicity and proves asymptotically a $30$-year old conjecture of Jackson and Ordaz.

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Nemanja Draganić, David Munhá Correia, Benny Sudakov. 2023-01-24. Chv\'atal-Erd\H{o}s condition for pancyclicity. https://arxiv.org/abs/2301.10190

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