$(r)$-Pancyclic, $(r)$-Bipancyclic and Oddly $(r)$-Bipancyclic Graphs

TL;DR

This paper classifies specific types of cyclic graphs with given vertex counts and limited edges, using computer search to identify all such graphs with particular cycle properties.

Contribution

It provides a comprehensive classification of $(r)$-pancyclic, $(r)$-bipancyclic, and oddly $(r)$-bipancyclic graphs with constrained edges, expanding understanding of their structure.

Findings

Classified all $(r)$-pancyclic graphs with up to $v+5$ edges.

Classified all $(r)$-bipancyclic graphs with up to $v+5$ edges.

Classified all oddly $(r)$-bipancyclic graphs with up to $v+4$ edges.

Abstract

A graph with $v$ vertices is $(r)$-pancyclic if it contains precisely $r$ cycles of every length from 3 to $v$. A bipartite graph with even number of vertices $v$ is said to be $(r)$-bipancyclic if it contains precisely $r$ cycles of each even length from 4 to $v$. A bipartite graph with odd number of vertices $v$ and minimum degree at least 2 is said to be oddly $(r)$-bipancyclic if it contains precisely $r$ cycles of each even length from 4 to $v-1$. In this paper, using computer search, we classify all $(r)$-pancyclic and $(r)$-bipancyclic graphs with $v$ vertices and at most $v+5$ edges. We also classify all oddly $(r)$-bipancyclic graphs with $v$ vertices and at most $v+4$ edges.