Cycles and paths through specified vertices in graphs with a given clique number
Chengli Li, Leyou Xu
TL;DR
The paper generalizes Dirac-type results by investigating cycles and paths through high-degree vertices in $2$-connected graphs of order $n$ with fixed clique number $\omega$. It develops an ore-path framework and structural classifications (including the families $\mathcal{H}(n,\omega)$ and a specific exceptional graph) to prove that a cycle through all vertices of degree at least $n-\omega$ exists unless a precise obstruction occurs, and that a $(u,v)$-path through all vertices of degree at least $n-\omega+1$ exists unless similar obstructions arise. Additionally, the work yields a strengthened result showing pancyclicity under $\delta+\omega\ge n$ with explicit exceptional cases, and provides conditions for Hamiltonian-connectedness when $\delta+\omega\ge n+1$. These findings broaden Hamiltonicity and pancyclicity theory by tying vertex degree thresholds to clique structure and graph composition, with implications for cycle and path construction in graphs constrained by clique number.
Abstract
B. Bollobás and G. Brightwell and independently R. Shi proved the existence of a cycle through all vertices whose degrees at least $\frac{n}{2}$ in any $2$-connected graph of order $n$. Motivated by this result, we prove the existence of a cycle through all vertices whose degrees at least $n-ω$ in any $2$-connected graph $G$ of order $n$ with clique number $ω$ unless $G$ is a specific graph. Moreover, we show that for any pair of vertices whose degrees are at least $n-ω+1$ in a graph $G$ of order $n$ with clique number $ω$, there exists a path joining them which contains all vertices of degree at least $n-ω+1$ unless $G$ belongs to certain graph classes. In doing so, we prove the existence of a $(u,v)$-path through all vertices whose degrees at least $\frac{n+1}{2}$ in any graph of order $n$, where $u,v$ are two distinct vertices of degree at least $\frac{n+1}{2}$.