Odd spanning trees of a graph
Jingyu Zheng, Baoyindureng Wu
TL;DR
The paper addresses when graphs contain connected odd factors and odd spanning trees, and when complements contain odd spanning trees, focusing on graphs of even order. It employs the Nash-Williams–Tutte framework to construct connected odd factors from pairs of edge-disjoint spanning trees and uses case analyses to derive tight degree conditions for odd spanning trees, plus full classifications for split graphs and complements of triangle-free graphs. Key results include that every $4$-edge-connected graph of even order has a connected odd factor, and that a bound of $\delta(G)\ge \frac{n}{2}+1$ guarantees an odd spanning tree (with tightness shown via $K_{\frac{n}{2},\frac{n}{2}}$ when $n$ is multiple of $4$). The work also provides complete characterizations for split graphs and, for triangle-free graphs, precise conditions under which the complement admits an odd spanning tree, and it outlines several open problems with potential implications for graph complements and diameter-related properties.
Abstract
A graph $G=(V,E)$ is said to be odd (or even, resp.) if $d_G(v)$ is odd (or even, resp.) for any $v\in V$. Trivially, the order of an odd graph must be even. In this paper, we show that every 4-edge connected graph of even order has a connected odd factor. A spanning tree $T$ of $G$ is called a homeomorphically irreducible spanning tree (HIST by simply) if $T$ contains no vertex of degree two. Trivially, an odd spanning tree must be a HIST. In 1990, Albertson, Berman, Hutchinson, and Thomassen showed that every connected graph of order $n$ with $δ(G)\geq \min\{\frac n 2, 4\sqrt{2n}\}$ contains a HIST. We show that every complete bipartite graph with both parts being even has no odd spanning tree, thereby for any even integer $n$ divisible by 4, there exists a graph of order $n$ with the minimum degree $\frac n 2$ having no odd spanning tree. Furthermore, we show that every graph of order $n$ with $δ(G)\geq \frac n 2 +1$ has an odd spanning tree. We also characterize all split graphs having an odd spanning tree. As an application, for any graph $G$ with diameter at least 4, $\overline{G}$ has a spanning odd double star. Finally, we also give a necessary and sufficient condition for a triangle-free graph $G$ whose complement contains an odd spanning tree. A number of related open problems are proposed.