Cycles and paths through specified vertices in graphs with a given clique number

TL;DR

This paper generalizes classical results on cycles and paths through high-degree vertices in graphs, establishing new conditions related to clique number and degree thresholds for 2-connected graphs.

Contribution

It introduces new theorems on the existence of cycles and paths through vertices with degrees related to the clique number, extending prior degree-based results.

Findings

Existence of a cycle through vertices with degree at least n - ω in 2-connected graphs with clique number ω.

Existence of a path connecting vertices with degree at least n - ω + 1, passing through all such vertices.

Existence of a (u,v)-path through vertices with degree at least (n+1)/2 in any graph.

Abstract

B. Bollob\'{a}s 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 $n-\omega$ in any $2$-connected graph $G$ of order $n$ with clique number $\omega$ unless $G$ is a specific graph. Moreover, we show that for any pair of vertices whose degrees are at least $n-\omega+1$ in a graph $G$ of order $n$ with clique number $\omega$, there exists a path joining them which contains all vertices of degree at least $n-\omega+1$ 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 $\frac{n+1}{2}$.