Arboricity and transversal problems on bounded degree graphs

Abstract

In this thesis, we investigate arboricity and transversal problems on graphs in a bounded degree setting. We pay particular attention to problems involving both the maximum degree and maximum density of the graphs.

The first topic is bounded degree arboricity. The arboricity of a multigraph $G$ is the minimum number of forests required to cover its edge set. It is well understood through a theorem of Nash-Williams. For the problem of bounded degree arboricity, we seek to cover the edge set of a multigraph $G$ by the minimum number of forests such that every vertex $v$ has maximum degree at most $f(v)$ in every forest, for some fixed weight function $f : V(G) \rightarrow \mathbb{Z}_{\ge 2}$. The case $f = 2$ is referred to as linear arboricity. The Linear Arboricity Conjecture, due to Akiyama, Exoo, and Harary, asserts that the linear arboricity of a simple graph $G$ with maximum degree $\Delta$ is at most $\lceil (\Delta+1)/2 \rceil$. Using tools from orientations of multigraphs, we prove that the Linear Arboricity Conjecture holds for multigraphs whose maximum degree is sufficiently large compared to a certain maximum density. This improves on previous results of this type, and moreover our methods extend to give a general upper bound on the bounded degree arboricity of multigraphs, as a maximum of a weighted maximum degree and maximum density of the multigraph. In addition, we disprove a conjecture of Truszczy\'nski which proposes a more precise upper bound on the bounded degree arboricity of multigraphs, but we also show that the conjecture does hold for simple graphs with sufficiently large girth, and that it holds for all simple graphs asymptotically.

The second topic in this thesis is independent transversals in bounded degree graphs. Given a graph $G$ and a partition $\mathcal{P} = \{V_1, \ldots, V_n\}$ of its vertex set, an independent transversal is a transversal of $\mathcal{P}$ that is independent in $G$. Haxell proved that there exists an independent transversal when $G$ has maximum degree $\Delta$ and $|V_i| \ge 2\Delta$ for every $i$, and this condition has been shown to be best possible by Szab\'o and Tardos. Wanless and Wood proved the existence of an independent transversal when the maximum class average degree (a kind of maximum density) of $G, \mathcal{P}$ is $D$ and $|V_i| \ge 4D$ for every $i$, and this condition has been shown to be asymptotically best possible by Groenland, Kaiser, Treffers, and Wales. Using topological methods, we find an interpolation between these two sufficient conditions, and moreover we give a construction showing that our result is best possible. The method for our tight construction, based on a simple lemma, is then further explored. We use our method to streamline tight constructions for Haxell's theorem, and we prove that our method in fact yields all possible extremal constructions. We also use it to give more streamlined counterexamples to a list coloring conjecture of Reed. Then we adapt the construction method to the setting of full rainbow matchings of multi-hypergraphs, which are independent transversals in their line graphs. Aharoni, Berger, and Meshulam proved that there exists a full rainbow matching in an $r$-uniform multigraph $G$ with maximum degree $\Delta$ when every edge class has size $|E_i| \ge r\Delta$. We use our method to give tight constructions for this theorem. We also describe counterexamples to a color degree generalization of Galvin's Theorem on list edge-coloring bipartite multigraphs. Finally, we describe how our method applies to independent transversals in uniform hypergraphs and other related settings.