Forbidden Subgraphs for Chorded Pancyclicity

TL;DR

This paper investigates the conditions under which claw-free graphs are guaranteed to contain all possible cycle lengths with chords, by identifying specific forbidden subgraphs that ensure chorded pancyclicity.

Contribution

It introduces the concept of chorded pancyclicity and characterizes forbidden subgraphs in claw-free graphs that guarantee this property.

Findings

Certain paths and triangles with pendant paths are forbidden.

Forbidden subgraphs imply the existence of chorded cycles of all lengths.

The paper establishes sufficient conditions for chorded pancyclicity in claw-free graphs.

Abstract

We call a graph $G$ pancyclic if it contains at least one cycle of every possible length $m$, for $3\le m\le |V(G)|$. In this paper, we define a new property called chorded pancyclicity. We explore forbidden subgraphs in claw-free graphs sufficient to imply that the graph contains at least one chorded cycle of every possible length $4, 5, \ldots, |V(G)|$. In particular, certain paths and triangles with pendant paths are forbidden.