Cyclability of $id$-cycles in graphs

TL;DR

This paper introduces the concept of $id$-cycles in graphs, proving that every such cycle can be extended to a larger cycle, thereby generalizing existing results on Hamiltonicity and cyclability.

Contribution

It defines $id$-cycles based on implicit-degrees and proves their extendability to larger cycles, broadening the understanding of graph cyclability.

Findings

Every $id$-cycle is contained in a larger cycle in the graph.

The result generalizes previous theorems on Hamiltonian cycles.

Provides a new criterion for cycle extendability based on implicit-degrees.

Abstract

Let $G$ be a graph on $n$ vertices and $C'=v_0v_1\cdots v_{p-1}v_0$ a vertex sequence of $G$ with $p\geq 3$ ($v_i\neq v_j$ for all $i,j=0,1,\ldots,p-1$, $i\neq j$). If for any successive vertices $v_i$, $v_{i+1}$ on $C'$, either $v_iv_{i+1}\in E(G)$ or both of the first implicit-degrees of $v_i$ and $v_{i+1}$ are at least $n/2$ (indices are taken modulo $p$), then $C'$ is called an $id$-cycle of $G$. In this paper, we prove that for every $id$-cycle $C'$, there exists a cycle $C$ in $G$ with $V(C')\subseteq V(C)$. This generalizes several early results on the Hamiltonicity and cyclability of graphs.