On non-planar, cycle-conformal graphs

TL;DR

The paper advances the theory of cycle-conformal graphs within matching-covered graphs by providing complete characterisations for two key bipartite classes in the non-planar setting: (i) the unique brace that is Pfaffian and cycle-conformal is $C_4$, and (ii) the unique cubic brace that is cycle-conformal is $K_{3,3}$. Building on tight-cut decomposition, the authors show that in cubic bipartite graphs cycle-conformality reduces to the braces, and in Pfaffian bipartite graphs, cycle-conformality coincides with planarity. They further prove that a cubic, bipartite, matching-covered graph is cycle-conformal iff all its braces are cycle-conformal, and that a matching-covered, Pfaffian, bipartite, cycle-conformal graph is cycle-conformal precisely when it is planar. The authors conjecture that for each $\\ell \\geq 2$, the only $\\ell$-regular, cycle-conformal brace is $K_{\\ell,\\ell}$, with the case $\\ell=3$ resolved as $K_{3,3}$. These results sharpen our understanding of cycle-conformal graphs and have implications for perfect-matching enumeration, Pfaffian recognition, and related structural questions in matching theory.

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 $K_{\ell,\ell}$.

Figures (12)

• Figure 1: A matching covered, planar, bipartite graph whose braces are all isomorphic to $C_4$. In the graph in the middle a non-conformal even cycle is indicated in red. On the right a tight cut is indicated such that both tight cut contractions corresponding to it are cycle-conformal.
• Figure 2: A drawing of two copies of the Heawood graph. On the right a non-conformal, even cycle in the Heawood graph is marked in red.
• Figure 3: One of the graphs described in lem:counterexample with $\ell = 4$. The tight cut $F$ is indicated by the red dashed line and the problematic cycle $C$ is traced in blue.
• Figure 4: An illustration of the kind of $K_{3,3}$ we purport to exist in lem:k33subdivision. The five bold lines represent single edges and the dashed lines represent non-trivial paths.
• Figure 5: The figure to the left represents the situation in the proof of lem:k33subdivision before claim:nobouncing and the figure to the right represents the graph $H$ we can construct prior to claim:noskipping with the help of claim:nobouncing. In the left figure $Q_1$ and $Q_2$ are indicated by the orange paths. To the right, the green path is $Q_w$ and the orange path is $Q_u$.

Theorems & Definitions (54)

• Proposition 1.1: folklore
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Conjecture 1.5
• Proposition 1.6: Dalwadi, Pause, Diwan, and Kothari DalwadiPDK2025Planar
• Lemma 1.7
• Theorem 1.8
• Theorem 1.9
• Proposition 2.1: Menger Menger1927Zur