On cycle covers of infinite bipartite graphs

TL;DR

This work investigates Salia's cycle-cover conjecture within infinite bipartite graphs that satisfy the double Hall property, extending the problem from finite graphs to two infinite regimes. The authors develop infinite-graph tools, notably ends and $F$-limits of cycles, to analyze when a set $A$ can be covered by cycles or by a collection of disjoint $2$-regular subgraphs, under various finiteness restrictions on the sides $A$ and $B$. They establish partial positive results under local finiteness constraints (e.g., when $A$ is locally finite and $B$ has restricted neighborhoods, or when $B$ is locally finite with $A$ countable) and demonstrate counterexamples showing the conjecture fails in wide infinite settings. They further relate finite-case outcomes to Lavrov–Vandenbussche-type results, showing that satisfying the conjecture for finite graphs can imply related properties for infinite graphs under certain degree restrictions. Overall, the paper maps a landscape of when infinite double-Hall graphs admit A-covers by cycles or $2$-regular subgraphs, highlighting both obstructions and constructive conditions.

Abstract

Given a graph $G$ and a subset $X$ of vertices of $G$ with size at least two, we denote by $N^2_G(X)$ the set of vertices of $G$ that have at least two neighbors in $X$. We say that a bipartite graph $G$ with sides $A$ and $B$ satisfies the double Hall property if for every subset $X$ of vertices of $A$ with size at least 2, $\vert N^2_G(X)\vert \geq \vert X\vert$. Salia conjectured that if $G$ is a bipartite graph that satisfies the double Hall property, then there exists a cycle in $G$ that covers all vertices of $A$. In this work, we study this conjecture restricted to infinite graphs. For this, we use the definition of ends and infinite cycles. It is simple to see that Salia's conjecture is false for infinite graphs in general. Consequently, all our results are partial. Under certain hypothesis it is possible to obtain a collection of pairwise disjoint 2-regular subgraphs that covers $A$. We show that if side $B$ is locally finite and side $A$ is countable, then the conjecture is true. Furthermore, assuming the conjecture holds for finite graphs, we show that it holds for infinite graphs with a restriction on the degree of the vertices of $B$. This result is inspired by the result obtained by Barát, Grzesik, Jung, Nagy and Pálvölgyi for finite graphs. Finally, we also show that if Salia's conjecture holds for some cases of infinite graphs, then the conjecture about finite graphs presented by Lavrov and Vandenbussche is true.

Figures (7)

• Figure 1: $\Gamma$ (graph $H$)
• Figure 2: The graph $\Gamma$
• Figure 3: Counterexample to Salia's conjecture for infinite graphs
• Figure 4: On the left side, in red, are the edges $wu$ and $vu$ that were originally part of $C$. On the right side, in red, the process mentioned in the text has already been carried out. Now, the edges that are part of $C$ are $wu_1$, $u_1v_1$ and $v_1Tv_0$.
• Figure 5: On the left side, we have the subgraph $C$ after the first modification made in Case 1. On the right side, the second process mentioned in Case 1 was carried out. In red are the edges $vu_3$, $u_3v_3$, and $v_3Tv_2$, which are now part of the subgraph $C$.

Theorems & Definitions (52)

• Definition 1.1
• Conjecture 1.2: Salia's conjecture Salia
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10