Vertex Arboricity of Toroidal Graphs with a Forbidden Cycle

The vertex arboricity $a(G)$ of a graph $G$ is the minimum $k$ such that $V(G)$ can be partitioned into $k$ sets where each set induces a forest. For a planar graph $G$, it is known that $a(G)\leq 3$. In two recent papers, it was proved that planar graphs without $k$-cycles for some $k\in\{3, 4, 5, 6, 7\}$ have vertex arboricity at most 2. For a toroidal graph $G$, it is known that $a(G)\leq 4$. Let us consider the following question: do toroidal graphs without $k$-cycles have vertex arboricity at most 2? It was known that the question is true for k=3, and recently, Zhang proved the question is true for $k=5$. Since a complete graph on 5 vertices is a toroidal graph without any $k$-cycles for $k\geq 6$ and has vertex arboricity at least three, the only unknown case was k=4. We solve this case in the affirmative; namely, we show that toroidal graphs without 4-cycles have vertex arboricity at most 2.Comment: 8 pages, 2 figure

Planar graphs without 3-cycles and with 4-cycles far apart are 3-choosable

A graph G is said to be L-colourable if for a given list assignment L = {L(v)|v ∈ V (G)} there is a proper colouring c of G such that c(v) ∈ L(v) for all v in V (G). If G is L-colourable for all L with |L(v)| ≥ k for all v in V (G), then G is said to be k-choosable. This paper focuses on two different ways to prove list colouring results on planar graphs. The first method will be discharging, which will be used to fuse multiple results into one theorem. The second method will be restricting the lists of vertices on the boundary and applying induction, which will show that planar graphs without 3- cycles and 4-cycles distance 8 apart are 3-choosable

Edge-choosability of Planar Graphs

According to the List Colouring Conjecture, if G is a multigraph then χ' (G)=χl' (G) . In this thesis, we discuss a relaxed version of this conjecture that every simple graph G is edge-(∆ + 1)-choosable as by Vizing’s Theorem ∆(G) ≤χ' (G)≤∆(G) + 1. We prove that if G is a planar graph without 7-cycles with ∆(G)≠5,6 , or without adjacent 4-cycles with ∆(G)≠5, or with no 3-cycles adjacent to 5-cycles, then G is edge-(∆ + 1)-choosable

Box representations of embedded graphs

A $d$-box is the cartesian product of $d$ intervals of $\mathbb{R}$ and a $d$-box representation of a graph $G$ is a representation of $G$ as the intersection graph of a set of $d$-boxes in $\mathbb{R}^d$. It was proved by Thomassen in 1986 that every planar graph has a 3-box representation. In this paper we prove that every graph embedded in a fixed orientable surface, without short non-contractible cycles, has a 5-box representation. This directly implies that there is a function $f$, such that in every graph of genus $g$, a set of at most $f(g)$ vertices can be removed so that the resulting graph has a 5-box representation. We show that such a function $f$ can be made linear in $g$. Finally, we prove that for any proper minor-closed class $\mathcal{F}$, there is a constant $c(\mathcal{F})$ such that every graph of $\mathcal{F}$ without cycles of length less than $c(\mathcal{F})$ has a 3-box representation, which is best possible.Comment: 16 pages, 6 figures - revised versio

Spanning trees without adjacent vertices of degree 2

Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that there exist highly connected graphs in which every spanning tree contains vertices of degree 2. Using a result of Alon and Wormald, we show that there exists a natural number $d$ such that every graph of minimum degree at least $d$ contains a spanning tree without adjacent vertices of degree 2. Moreover, we prove that every graph with minimum degree at least 3 has a spanning tree without three consecutive vertices of degree 2

Circumference and Pathwidth of Highly Connected Graphs

Birmele [J. Graph Theory, 2003] proved that every graph with circumference t has treewidth at most t-1. Under the additional assumption of 2-connectivity, such graphs have bounded pathwidth, which is a qualitatively stronger result. Birmele's theorem was extended by Birmele, Bondy and Reed [Combinatorica, 2007] who showed that every graph without k disjoint cycles of length at least t has bounded treewidth (as a function of k and t). Our main result states that, under the additional assumption of (k + 1)- connectivity, such graphs have bounded pathwidth. In fact, they have pathwidth O(t^3 + tk^2). Moreover, examples show that (k + 1)-connectivity is required for bounded pathwidth to hold. These results suggest the following general question: for which values of k and graphs H does every k-connected H-minor-free graph have bounded pathwidth? We discuss this question and provide a few observations.Comment: 11 pages, 4 figure