2 research outputs found
4-Factor-criticality of vertex-transitive graphs
A graph of order is -factor-critical, where is an integer of the
same parity as , if the removal of any set of 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 graphs of even order and degree at least 5 is
2-extendable.Comment: 34 pages, 3 figure
Small separations in vertex transitive graphs
Let be an integer. We prove a rough structure theorem for separations of
order at most in finite and infinite vertex transitive graphs. Let be a vertex transitive graph, let be a finite vertex-set
with and |\{v \in V \setminus A : {u \sim vu \in
A} \}|\le k. We show that whenever the diameter of is at least
, either , or has a ring-like structure (with
bounded parameters), and is efficiently contained in an interval. This
theorem may be viewed as a rough characterization, generalizing an earlier
result of Tindell, and has applications to the study of product sets and
expansion in groups.Comment: 28 page