Vertex arboricity of triangle-free graphs

Master's Project (M.S.) University of Alaska Fairbanks, 2016The vertex arboricity of a graph is the minimum number of colors needed to color the vertices so that the subgraph induced by each color class is a forest. In other words, the vertex arboricity of a graph is the fewest number of colors required in order to color a graph such that every cycle has at least two colors. Although not standard, we will refer to vertex arboricity simply as arboricity. In this paper, we discuss properties of chromatic number and k-defective chromatic number and how those properties relate to the arboricity of trianglefree graphs. In particular, we find bounds on the minimum order of a graph having arboricity three. Equivalently, we consider the largest possible vertex arboricity of triangle-free graphs of fixed order

The degree-diameter problem for sparse graph classes

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is $\Theta(\Delta^{k-1})$, and for graphs of bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases for fixed $k$. For graphs of given treewidth, we determine the the maximum number of vertices up to a constant factor. More precise bounds are given for graphs of given treewidth, graphs embeddable on a given surface, and apex-minor-free graphs

Graph Treewidth and Geometric Thickness Parameters

Consider a drawing of a graph $G$ in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of $G$, is the classical graph parameter "thickness". By restricting the edges to be straight, we obtain the "geometric thickness". By further restricting the vertices to be in convex position, we obtain the "book thickness". This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth $k$, the maximum thickness and the maximum geometric thickness both equal $\lceil{k/2}\rceil$. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth $k$, the maximum book thickness equals $k$ if $k \leq 2$ and equals $k+1$ if $k \geq 3$. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in Computer Science 3843:129-140, Springer, 2006. The full version was published in Discrete & Computational Geometry 37(4):641-670, 2007. That version contained a false conjecture, which is corrected on page 26 of this versio

Testing bounded arboricity

In this paper we consider the problem of testing whether a graph has bounded arboricity. The family of graphs with bounded arboricity includes, among others, bounded-degree graphs, all minor-closed graph classes (e.g. planar graphs, graphs with bounded treewidth) and randomly generated preferential attachment graphs. Graphs with bounded arboricity have been studied extensively in the past, in particular since for many problems they allow for much more efficient algorithms and/or better approximation ratios. We present a tolerant tester in the sparse-graphs model. The sparse-graphs model allows access to degree queries and neighbor queries, and the distance is defined with respect to the actual number of edges. More specifically, our algorithm distinguishes between graphs that are $\epsilon$-close to having arboricity $\alpha$ and graphs that $c \cdot \epsilon$-far from having arboricity $3\alpha$, where $c$ is an absolute small constant. The query complexity and running time of the algorithm are $\tilde{O}\left(\frac{n}{\sqrt{m}}\cdot \frac{\log(1/\epsilon)}{\epsilon} + \frac{n\cdot \alpha}{m} \cdot \left(\frac{1}{\epsilon}\right)^{O(\log(1/\epsilon))}\right)$ where $n$ denotes the number of vertices and $m$ denotes the number of edges. In terms of the dependence on $n$ and $m$ this bound is optimal up to poly-logarithmic factors since $\Omega(n/\sqrt{m})$ queries are necessary (and $\alpha = O(\sqrt{m}))$. We leave it as an open question whether the dependence on $1/\epsilon$ can be improved from quasi-polynomial to polynomial. Our techniques include an efficient local simulation for approximating the outcome of a global (almost) forest-decomposition algorithm as well as a tailored procedure of edge sampling

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