On non-planar, cycle-conformal graphs

TL;DR

This paper characterizes non-planar, bipartite, and cubic cycle-conformal graphs, extending previous planar results, and identifies key structural building blocks like $K_{3,3}$ and $C_4$.

Contribution

It provides the first complete characterization of non-planar, cycle-conformal, bipartite, and Pfaffian graphs, identifying fundamental building blocks and proposing a conjecture for regular braces.

Findings

$C_4$ is the only Pfaffian, cycle-conformal brace.

$K_{3,3}$ is the only cubic, cycle-conformal brace.

Characterizations facilitate understanding of matching covered graphs.

Abstract

A graph $G$ is called matching covered if all of its edges are contained in some perfect matching of $G$. Furthermore, a cycle $C \subseteq G$ is called conformal if $G - V(C)$ has a perfect matching and $G$ itself is called cycle-conformal if all of its even cycles are conformal. Both matching covered graphs and conformal cycles play central roles in matching theory. After a string of results from various authors, focused mainly on bipartite, planar graphs and claw-free graphs, a complete characterisation of all planar, cycle-conformal graphs has recently been presented by Dalwadi, Pause, Diwan, and Kothari [DMTCS, 2025]. We continue this exploration further into the realm of non-planar graphs, giving a characterisation of matching covered, cycle-conformal graphs that are bipartite and cubic, and respectively, those that are bipartite and Pfaffian. The last class plays a fundamental role in matching theory, having important connections to the problem of counting perfect matchings, recognising graphs with even directed cycles, and computing the permanent of certain matrices efficiently. To prove our results, we break matching covered graphs down to their building blocks, the bipartite ones of which are called braces. The key to both characterisations are theorems that identify the braces in the respective classes. In particular, as our main results, we show that the cycle of length 4 is the only Pfaffian, cycle-conformal brace and we show that $K_{3,3}$ is the only cubic, cycle-conformal brace. In both cases these theorems facilitate the characterisations of the much richer classes of associated matching covered graphs. We conjecture that for each integer $\ell \geq 2$ the only $\ell$-regular, cycle-conformal brace is $K_{\ell,\ell}$.