1,607 research outputs found
3-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 factor-critical graphs and bicritical graphs, respectively. It is
well known that every connected vertex-transitive graph of odd order is
factor-critical and every connected non-bipartite vertex-transitive graph of
even order is bicritical. In this paper, we show that a simple connected
vertex-transitive graph of odd order at least 5 is 3-factor-critical if and
only if it is not a cycle.Comment: 15 pages, 3 figure
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
The mincut graph of a graph
In this paper we introduce an intersection graph of a graph , with vertex
set the minimum edge-cuts of . We find the minimum cut-set graphs of some
well-known families of graphs and show that every graph is a minimum cut-set
graph, henceforth called a \emph{mincut graph}. Furthermore, we show that
non-isomorphic graphs can have isomorphic mincut graphs and ask the question
whether there are sufficient conditions for two graphs to have isomorphic
mincut graphs. We introduce the -intersection number of a graph , the
smallest number of elements we need in in order to have a family of subsets, such that for each subset. Finally we
investigate the effect of certain graph operations on the mincut graphs of some
families of graphs
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