Every $2$-connected $[4, 2]$-graph of order at least seven contains a pancyclic edge

TL;DR

This paper investigates when graphs necessarily contain a pancyclic edge, focusing on $2$-connected $[4,2]$-graphs of order at least $7$. The authors prove that such graphs always contain a pancyclic edge, structuring the proof around a key lemma that guarantees the existence of a $igl\{3,4,5\bigr\}$-cyclic edge. They also determine extremal sizes for $[4,2]$-graphs of order $n$, with equality cases such as $K_{loor{n/2}}+K_{loor{n/2}}$, $B_n$, and $B_n^+$, and prove that $[4,2]$-graphs of order at least $8$ are not uniquely Hamiltonian. Together, these results advance the understanding of pancyclic phenomena in this graph class and connect to broader questions about pancyclicity and Hamiltonicity in graphs.

Abstract

A graph $G$ is called an $[s,t]$-graph if any induced subgraph of $G$ of order $s$ has size at least $t.$ An edge $e$ in a graph $G$ of order $n$ is called pancyclic if for every integer $k$ with $3\le k\le n,$ $e$ lies in a $k$-cycle. We prove that every $2$-connected $[4, 2]$-graph of order at least seven contains a pancyclic edge. This strengthens an existing result. We also determine the minimum size of a $[4, 2]$-graph of a given order and show that any $[4, 2]$-graph of order at least eight is not uniquely hamiltonian.

Figures (4)

• Figure 1: The diamond and house
• Figure 2: A uniquely hamiltonian $[4,2]$-graph of order seven
• Figure 3: The graph $BT(7)$
• Figure 4: A vertex-pancyclic graph that contains no pancyclic edge