Zhan’s pancyclicity conjecture for 2-connected [p+2,p]-graphs when p=4 or 5

Determine whether every 2-connected [p+2,p]-graph of order at least 2p+3 with minimum degree at least p is pancyclic for each p\in\{4,5\}, thereby resolving the cases of Zhan’s conjecture not settled by the present paper.

Background

An [s,t]-graph is a graph in which every induced subgraph on s vertices has at least t edges. Zhan conjectured that, for p\in{3,4,5}, every 2-connected [p+2,p]-graph with at least 2p+3 vertices and minimum degree at least p is pancyclic, meaning that it contains cycles of every length from 3 through its order.

The paper proves the conjecture for p=3 by showing that every 2-connected [5,3]-graph of order at least 9 with minimum degree at least 3 is pancyclic. Consequently, the unresolved portions explicitly identified by the conjecture’s scope are the cases p=4 and p=5: 2-connected [6,4]-graphs of order at least 11 with minimum degree at least 4, and 2-connected [7,5]-graphs of order at least 13 with minimum degree at least 5, respectively.

References

In 2024, Zhan conjectured that every $2$-connected $[p + 2, p]$-graph of order at least $2p + 3$ and with minimum degree at least $p$ is pancyclic, where $p$ is an integer with $3 leq p leq 5$.

— On the pancyclicity of $2$-connected $[5,3]$-graphs  (2509.20038 - Liu et al., 24 Sep 2025) in Abstract; Section 1, Introduction, immediately following the definition of [s,t]-graphs