1,771 research outputs found
An approximate version of Sidorenko's conjecture
A beautiful conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if H
is a bipartite graph, then the random graph with edge density p has in
expectation asymptotically the minimum number of copies of H over all graphs of
the same order and edge density. This conjecture also has an equivalent
analytic form and has connections to a broad range of topics, such as matrix
theory, Markov chains, graph limits, and quasirandomness. Here we prove the
conjecture if H has a vertex complete to the other part, and deduce an
approximate version of the conjecture for all H. Furthermore, for a large class
of bipartite graphs, we prove a stronger stability result which answers a
question of Chung, Graham, and Wilson on quasirandomness for these graphs.Comment: 12 page
Upper bounds on the k-forcing number of a graph
Given a simple undirected graph and a positive integer , the
-forcing number of , denoted , is the minimum number of vertices
that need to be initially colored so that all vertices eventually become
colored during the discrete dynamical process described by the following rule.
Starting from an initial set of colored vertices and stopping when all vertices
are colored: if a colored vertex has at most non-colored neighbors, then
each of its non-colored neighbors becomes colored. When , this is
equivalent to the zero forcing number, usually denoted with , a recently
introduced invariant that gives an upper bound on the maximum nullity of a
graph. In this paper, we give several upper bounds on the -forcing number.
Notable among these, we show that if is a graph with order and
maximum degree , then . This simplifies to, for the zero forcing number case
of , . Moreover, when and the graph is -connected, we prove that , which is an improvement when , and
specializes to, for the zero forcing number case, . These results resolve a problem posed by
Meyer about regular bipartite circulant graphs. Finally, we present a
relationship between the -forcing number and the connected -domination
number. As a corollary, we find that the sum of the zero forcing number and
connected domination number is at most the order for connected graphs.Comment: 15 pages, 0 figure
The maximum forcing number of polyomino
The forcing number of a perfect matching of a graph is the
cardinality of the smallest subset of that is contained in no other perfect
matchings of . For a planar embedding of a 2-connected bipartite planar
graph which has a perfect matching, the concept of Clar number of hexagonal
system had been extended by Abeledo and Atkinson as follows: a spanning
subgraph of is called a Clar cover of if each of its components is
either an even face or an edge, the maximum number of even faces in Clar covers
of is called Clar number of , and the Clar cover with the maximum number
of even faces is called the maximum Clar cover. It was proved that if is a
hexagonal system with a perfect matching and is a set of hexagons in a
maximum Clar cover of , then has a unique 1-factor. Using this
result, Xu {\it et. at.} proved that the maximum forcing number of the
elementary hexagonal system are equal to their Clar numbers, and then the
maximum forcing number of the elementary hexagonal system can be computed in
polynomial time. In this paper, we show that an elementary polyomino has a
unique perfect matching when removing the set of tetragons from its maximum
Clar cover. Thus the maximum forcing number of elementary polyomino equals to
its Clar number and can be computed in polynomial time. Also, we have extended
our result to the non-elementary polyomino and hexagonal system
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