Recent trends and future directions in vertex-transitive graphs

A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade

Extremal Directed And Mixed Graphs

We consider three problems in extremal graph theory, namely the degree/diameter problem, the degree/geodecity problem and Tur\'{a}n problems, in the context of directed and partially directed graphs. A directed graph or mixed graph $G$ is $k$-geodetic if there is no pair of vertices $u,v$ of $G$ such that there exist distinct non-backtracking walks with length $\leq k$ in $G$ from $u$ to $v$. The order of a $k$-geodetic digraph with minimum out-degree $d$ is bounded below by the \emph{directed Moore bound} $M(d,k) = 1+d+d^2+\dots +d^k$; similarly the order of a $k$-geodetic mixed graph with minimum undirected degree $r$ and minimum directed out-degree $z$ is bounded below by the \emph{mixed Moore bound}. We will be interested in networks with order exceeding the Moore bound by some small \emph{excess} $\epsilon$. The \emph{degree/geodecity problem} asks for the smallest possible order of a $k$-geodetic digraph or mixed graph with given degree parameters. We prove the existence of extremal graphs, which we call \emph{geodetic cages}, and provide some bounds on their order and information on their structure. We discuss the structure of digraphs with excess one and rule out the existence of certain digraphs with excess one. We then classify all digraphs with out-degree two and excess two, as well as all diregular digraphs with out-degree two and excess three. We also present the first known non-trivial examples of geodetic cages. We then generalise this work to the setting of mixed graphs. First we address the question of the total regularity of mixed graphs with order close to the Moore bound and prove bounds on the order of mixed graphs that are not totally regular. In particular using spectral methods we prove a conjecture of L\'{o}pez and Miret that mixed graphs with diameter two and order one less than the Moore bound are totally regular. Using counting arguments we then provide strong bounds on the order of totally regular $k$-geodetic mixed graphs and use these results to derive new extremal mixed graphs. Finally we change our focus and study the Tur\'{a}n problem of the largest possible size of a $k$-geodetic digraph with given order. We solve this problem and also prove an exact expression for the restricted problem of the largest possible size of strongly connected $2$-geodetic digraphs, as well as providing constructions of strongly connected $k$-geodetic digraphs that we conjecture to be extremal for larger $k$. We close with a discussion of some related generalised Tur\'{a}n problems for $k$-geodetic digraphs