Choosability of Graphs with Bounded Order: Ohba's Conjecture and Beyond

The \emph{choice number} of a graph $G$, denoted $\ch(G)$, is the minimum integer $k$ such that for any assignment of lists of size $k$ to the vertices of $G$, there is a proper colouring of $G$ such that every vertex is mapped to a colour in its list. For general graphs, the choice number is not bounded above by a function of the chromatic number. In this thesis, we prove a conjecture of Ohba which asserts that $\ch(G)=\chi(G)$ whenever $|V(G)|\leq 2\chi(G)+1$. We also prove a strengthening of Ohba's Conjecture which is best possible for graphs on at most $3\chi(G)$ vertices, and pose several conjectures related to our work.Comment: Master's Thesis, McGill Universit

The strong chromatic index of 1-planar graphs

The chromatic index $\chi'(G)$ of a graph $G$ is the smallest $k$ for which $G$ admits an edge $k$-coloring such that any two adjacent edges have distinct colors. The strong chromatic index $\chi'_s(G)$ of $G$ is the smallest $k$ such that $G$ has a proper edge $k$-coloring with the condition that any two edges at distance at most 2 receive distinct colors. A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, we show that every graph $G$ with maximum average degree $\bar{d}(G)$ has $\chi'_{s}(G)\le (2\bar{d}(G)-1)\chi'(G)$. As a corollary, we prove that every 1-planar graph $G$ with maximum degree $\Delta$ has $\chi'_{\rm s}(G)\le 14\Delta$, which improves a result, due to Bensmail et al., which says that $\chi'_{\rm s}(G)\le 24\Delta$ if $\Delta\ge 56$

On the complexity of some colorful problems parameterized by treewidth

In this paper,we study the complexity of several coloring problems on graphs, parameterizedby the treewidth of the graph.1. The List Coloring problem takes as input a graph G, togetherwith an assignment to each vertex v of a set of colors Cv. The problem is to determinewhether it is possible to choose a color for vertex v from the set of permitted colors Cv, for each vertex, so that the obtained coloring of G is proper. We show that this problem is W[1]-hard, parameterized by the treewidth of G. The closely related Precoloring Extension problem is also shown to be W[1]-hard, parameterized by treewidth.2. An equitable coloring of a graph G is a proper coloring of the verticeswhere the numbers of vertices having any two distinct colors differs by at most one.We show that the problem is hard forW[1], parameterized by the treewidth plus the number of colors.We also show that a list-based variation, List Equitable Coloring is W[1]-hard for forests, parameterizedby the number of colors on the lists.3. The list chromatic number χl(G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list Lv of colors, where each list has length at least r, there is a choice of one color fromeach vertex list Lv yielding a proper coloring of G. We show that the problem of determining whether χl(G) ≤ r, the ListChromatic Number problem, is solvable in linear time on graphs of constant treewidth

On Arrangements of Orthogonal Circles

In this paper, we study arrangements of orthogonal circles, that is, arrangements of circles where every pair of circles must either be disjoint or intersect at a right angle. Using geometric arguments, we show that such arrangements have only a linear number of faces. This implies that orthogonal circle intersection graphs have only a linear number of edges. When we restrict ourselves to orthogonal unit circles, the resulting class of intersection graphs is a subclass of penny graphs (that is, contact graphs of unit circles). We show that, similarly to penny graphs, it is NP-hard to recognize orthogonal unit circle intersection graphs.Comment: Appears in the Proceedings of the 27th International Symposium on Graph Drawing and Network Visualization (GD 2019

Problems in Extremal Graph Theory

This dissertation consists of six chapters concerning a variety of topics in extremal graph theory.Chapter 1 is dedicated to the results in the papers with Antnio Giro, Gbor Mszros, and Richard Snyder. We say that a graph is path-pairable if for any pairing of its vertices there exist edge disjoint paths joining the vertices in eachpair. We study the extremal behavior of maximum degree and diameter in some classes of path-pairable graphs. In particular we show that a path-pairable planar graph must have a vertex of linear degree.In Chapter 2 we present a joint work with Antnio Giro and Teeradej Kittipassorn. Given graphs G and H, we say that a graph F is H-saturated in G if F is H-free subgraph of G, but addition of any edge from E(G) to F creates a copy of H. Here we deal with the case when G is a complete k-partite graph with n vertices in each class, and H is a complete graph on r vertices. We prove bounds, which are tight for infinitely many values of k and r, on the minimal number of edges in a H-saturated graph in G, for this choice of G and H, answering questions of Ferrara, Jacobson, Pfender, and Wenger; and generalizing a result of Roberts.Chapter 3 is about a joint paper with Antnio Giro and Teeradej Kittipassorn. A coloring of the vertices of a digraph D is called majority coloring if no vertex of D receives the same color as more than half of its outneighbours. This was introduced by van der Zypen in 2016. Recently, Kreutzer, Oum, Seymour, van der Zypen, and Wood posed a number of problems related to this notion: here we solve several of them.In Chapter 4 we present a joint work with Antnio Giro. We show that given any integer k there exist functions g1(k), g2(k) such that the following holds. For any graph G with maximum degree one can remove fewer than g1(k) ^{1/2} vertices from G so that the remaining graph H has k vertices of the same degree at least (H) g2(k). It is an approximate version of conjecture of Caro and Yuster; and Caro, Lauri, and Zarb, who conjectured that g2(k) = 0.Chapter 5 concerns results obtained together with Kazuhiro Nomoto, Julian Sahasrabudhe, and Richard Snyder. We study a graph parameter, the graph burning number, which is supposed to measure the speed of the spread of contagion in a graph. We prove tight bounds on the graph burning number of some classes of graphs and make a progress towards a conjecture of Bonato, Janssen, and Roshanbin about the upper bound of graph burning number of connected graphs.In Chapter 6 we present a joint work with Teeradej Kittipassorn. We study the set of possible numbers of triangles a graph on a given number of vertices can have. Among other results, we determine the asymptotic behavior of the smallest positive integer m such that there is no graph on n vertices with exactly m copies of a triangle. We also prove similar results when we replace triangle by any fixed connected graph

A Unified Approach to Distance-Two Colouring of Graphs on Surfaces

In this paper we introduce the notion of $\Sigma$-colouring of a graph $G$: For given subsets $\Sigma(v)$ of neighbours of $v$, for every $v\in V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that appear together in some $\Sigma(v)$ receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of graphs embedded in a surface. We prove a general result for graphs embeddable in a fixed surface, which implies asymptotic versions of Wegner's and Borodin's Conjecture on the planar version of these two colourings. Using a recent approach of Havet et al., we reduce the problem to edge-colouring of multigraphs, and then use Kahn's result that the list chromatic index is close to the fractional chromatic index. Our results are based on a strong structural lemma for graphs embeddable in a fixed surface, which also implies that the size of a clique in the square of a graph of maximum degree $\Delta$ embeddable in some fixed surface is at most $\frac32\,\Delta$ plus a constant.Comment: 36 page

Labeling of graphs, sumset of squares of units modulo n and resonance varieties of matroids

This thesis investigates problems in a number of different areas of graph theory and its applications in other areas of mathematics. Motivated by the 1-2-3-Conjecture, we consider the closed distinguishing number of a graph G, denoted by dis[G]. We provide new upper bounds for dis[G] by using the Combinatorial Nullstellensatz. We prove that it is NP-complete to decide for a given planar subcubic graph G, whether dis[G] = 2. We show that for each integer t there is a bipartite graph G such that dis[G] \u3e t. Then some polynomial time algorithms and NP-hardness results for the problem of partitioning the edges of a graph into regular and/or locally irregular subgraphs are presented. We then move on to consider Johnson graphs to find resonance varieties of some classes of sparse paving matroids. The last application we consider is in number theory, where we find the number of solutions of the equation x21 + _ _ _ + x2 k = c, where c 2 Zn, and xi are all units in the ring Zn. Our approach is combinatorial using spectral graph theory