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

Uniquely circular colourable and uniquely fractional colourable graphs of large girth

Given any rational numbers $r \geq r' >2$ and an integer $g$, we prove that there is a graph $G$ of girth at least $g$, which is uniquely $r$-colourable and uniquely $r'$-fractional colourable

The Chromatic Structure of Dense Graphs

This thesis focusses on extremal graph theory, the study of how local constraints on a graph affect its macroscopic structure. We primarily consider the chromatic structure: whether a graph has or is close to having some (low) chromatic number. Chapter 2 is the slight exception. We consider an induced version of the classical Turán problem. Introduced by Loh, Tait, Timmons, and Zhou, the induced Turán number ex(n, {H, F-ind}) is the greatest number of edges in an n-vertex graph with no copy of H and no induced copy of F. We asymptotically determine ex(n, {H, F-ind}) for H not bipartite and F neither an independent set nor a complete bipartite graph. We also improve the upper bound for ex(n, {H, K_{2, t}-ind}) as well as the lower bound for the clique number of graphs that have some fixed edge density and no induced K_{2, t}. The next three chapters form the heart of the thesis. Chapters 3 and 4 consider the Erdős-Simonovits question for locally r-colourable graphs: what are the structure and chromatic number of graphs with large minimum degree and where every neighbourhood is r-colourable? Chapter 3 deals with the locally bipartite case and Chapter 4 with the general case. While the subject of Chapters 3 and 4 is a natural local to global colouring question, it is also essential for determining the minimum degree stability of H-free graphs, the focus of Chapter 5. Given a graph H of chromatic number r + 1, this asks for the minimum degree that guarantees that an H-free graph is close to r-partite. This is analogous to the classical edge stability of Erdős and Simonovits. We also consider the question for the family of graphs to which H is not homomorphic, showing that it has the same answer. Chapter 6 considers sparse analogues of the results of Chapters 3 to 5 obtaining the thresholds at which the sparse problem degenerates away from the dense one. Finally, Chapter 7 considers a chromatic Ramsey problem first posed by Erdős: what is the greatest chromatic number of a triangle-free graph on $n$ vertices or with m edges? We improve the best known bounds and obtain tight (up to a constant factor) bounds for the list chromatic number, answering a question of Cames van Batenburg, de Joannis de Verclos, Kang, and Pirot

Local resilience of spanning subgraphs in sparse random graphs

For each real γ>0γ>0 and integers Δ≥2Δ≥2 and k≥1k≥1, we prove that there exist constants β>0β>0 and C>0C>0 such that for all p≥C(log⁡n/n)1/Δp≥C(log⁡n/n)1/Δ the random graph G(n,p)G(n,p) asymptotically almost surely contains – even after an adversary deletes an arbitrary (1/k−γ1/k−γ)-fraction of the edges at every vertex – a copy of every n-vertex graph with maximum degree at most Δ, bandwidth at most βn and at least Cmax⁡{p−2,p−1log⁡n}Cmax⁡{p−2,p−1log⁡n} vertices not in triangles

The dynamics of proving uncolourability of large random graphs I. Symmetric Colouring Heuristic

We study the dynamics of a backtracking procedure capable of proving uncolourability of graphs, and calculate its average running time T for sparse random graphs, as a function of the average degree c and the number of vertices N. The analysis is carried out by mapping the history of the search process onto an out-of-equilibrium (multi-dimensional) surface growth problem. The growth exponent of the average running time is quantitatively predicted, in agreement with simulations.Comment: 5 figure