Presentations for vertex-transitive graphs

We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex-transitive graph.

Two-Distance-Primitive Graphs

A 2-distance-primitive graph is a vertex-transitive graph whose vertex stabilizer is primitive on both the first step and the second step neighborhoods. Let $\Gamma$ be such a graph. This paper shows that either $\Gamma$ is a cyclic graph, or $\Gamma$ is a complete bipartite graph, or $\Gamma$ has girth at most $4$ and the vertex stabilizer acts faithfully on both the first step and the second step neighborhoods. Also a complete classification is given of such graphs satisfying that the vertex stabilizer acts $2$-transitively on the second step neighborhood. Finally, we determine the unique 2-distance-primitive graph which is locally cyclic.

Cayley Graphs Without a Bounded Eigenbasis

Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon–Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor’s result to nonabelian groups.

A survey on Hamiltonicity in Cayley graphs and digraphs on different groups

Lovász had posed a question stating whether every connected, vertex-transitive graph has a Hamilton path in 1969. There is a growing interest in solving this longstanding problem and still it remains widely open. In fact, it was known that only five vertex-transitive graphs exist without a Hamiltonian cycle which do not belong to Cayley graphs. A Cayley graph is the subclass of vertex-transitive graph, and in view of the Lovász conjecture, the attention has focused more toward the Hamiltonicity of Cayley graphs. This survey will describe the current status of the search for Hamiltonian cycles and paths in Cayley graphs and digraphs on different groups, and discuss the future direction regarding famous conjecture.

Forbidden Pairs with a Common Graph Generating Almost the Same Sets

Let $\mathcal{H}$ be a family of connected graphs. A graph $G$ is said to be $\mathcal{H}$-free if $G$ does not contain any members of $\mathcal{H}$ as an induced subgraph. Let $\mathcal{F}(\mathcal{H})$ be the family of connected $\mathcal{H}$-free graphs. In this context, the members of $\mathcal{H}$ are called forbidden subgraphs.In this paper, we focus on two pairs of forbidden subgraphs containing a common graph, and compare the classes of graphs satisfying each of the two forbidden subgraph conditions. Our main result is the following: Let $H_{1},H_{2},H_{3}$ be connected graphs of order at least three, and suppose that $H_{1}$ is twin-less. If the symmetric difference of $\mathcal{F}(\{H_{1},H_{2}\})$ and $\mathcal{F}(\{H_{1},H_{3}\})$ is finite and the tuple $(H_{1};H_{2},H_{3})$ is non-trivial in a sense, then $H_{2}$ and $H_{3}$ are obtained from the same vertex-transitive graph by successively replacing a vertex with a clique and joining the neighbors of the original vertex and the clique. Furthermore, we refine a result in [Combin. Probab. Comput. 22 (2013) 733–748] concerning forbidden pairs.

4-Factor-Criticality of Vertex-Transitive Graphs

A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-factor-critical graphs and 2-factor-critical graphs are well-known factor-critical graphs and bicritical graphs, respectively. It is known that if a connected vertex-transitive graph has odd order, then it is factor-critical, otherwise it is elementary bipartite or bicritical. In this paper, we show that a connected vertex-transitive non-bipartite graph of even order at least 6 is 4-factor-critical if and only if its degree is at least 5. This result implies that each connected non-bipartite Cayley graph of even order and degree at least 5 is 2-extendable.

A Note on Coloring Vertex-Transitive Graphs

We prove bounds on the chromatic number $\chi$ of a vertex-transitive graph in terms of its clique number $\omega$ and maximum degree $\Delta$. We conjecture that every vertex-transitive graph satisfies $\chi \le \max \{\omega, \left\lceil\frac{5\Delta + 3}{6}\right\rceil\}$, and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with $\Delta \ge 13$ we prove the Borodin–Kostochka conjecture, i.e., $\chi\le\max\{\omega,\Delta-1\}$.

Semiregular elements in cubic vertex-transitive graphs and the restricted Burnside problem

In this paper, we prove that the maximal order of a semiregular element in the automorphism group of a cubic vertex-transitive graph Γ does not tend to infinity as the number of vertices of Γ tends to infinity. This gives a solution (in the negative) to a conjecture of Peter Cameron, John Sheehan and the author [4, conjecture 2].However, with an application of the positive solution of the restricted Burnside problem, we show that this conjecture holds true when Γ is either a Cayley graph or an arc-transitive graph.

Cores of Vertex Transitive Graphs

A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets each of which induce a copy of its core? We show that normal Cayley graphs and vertex transitive graphs with cores half their size always admit such partitions. We also show that the vertex sets of vertex transitive graphs with cores less than half their size do not, in general, have such partitions.