109 research outputs found
Hamiltonicity, independence number, and pancyclicity
A graph on n vertices is called pancyclic if it contains a cycle of length l
for all 3 \le l \le n. In 1972, Erdos proved that if G is a Hamiltonian graph
on n > 4k^4 vertices with independence number k, then G is pancyclic. He then
suggested that n = \Omega(k^2) should already be enough to guarantee
pancyclicity. Improving on his and some other later results, we prove that
there exists a constant c such that n > ck^{7/3} suffices
The Cycle Spectrum of Claw-free Hamiltonian Graphs
If is a claw-free hamiltonian graph of order and maximum degree
with , then has cycles of at least many different lengths.Comment: 9 page
Cycles in the burnt pancake graphs
The pancake graph is the Cayley graph of the symmetric group on
elements generated by prefix reversals. has been shown to have
properties that makes it a useful network scheme for parallel processors. For
example, it is -regular, vertex-transitive, and one can embed cycles in
it of length with . The burnt pancake graph ,
which is the Cayley graph of the group of signed permutations using
prefix reversals as generators, has similar properties. Indeed, is
-regular and vertex-transitive. In this paper, we show that has every
cycle of length with . The proof given is a
constructive one that utilizes the recursive structure of . We also
present a complete characterization of all the -cycles in for , which are the smallest cycles embeddable in , by presenting their
canonical forms as products of the prefix reversal generators.Comment: Added a reference, clarified some definitions, fixed some typos. 42
pages, 9 figures, 20 pages of appendice
A Survey of Best Monotone Degree Conditions for Graph Properties
We survey sufficient degree conditions, for a variety of graph properties,
that are best possible in the same sense that Chvatal's well-known degree
condition for hamiltonicity is best possible.Comment: 25 page
A new approach to pancyclicity of Paley graphs I
Let be an undirected graph of order and let be an -cycle
graph. is called pancyclic if contains a for any . We show that the pancyclicity of specific Cayley graphs and
the Cartesian product of specific two graphs. As a corollary of these two
theorems, we provide a new proof of the pancyclicity of the Paley graph.Comment: Corrected the formatting of the references. Corrected the "Cref"
behavior, which is latex command. Add another proof of the pancyclicity of
Paley graph, which is already known. Change the title. Add a reference to
generalized Paley graph and my next paper(in preparation) Change the titl
An asymptotically tight bound on the Q-index of graphs with forbidden cycles
Let G be a graph of order n and let q(G) be that largest eigenvalue of the
signless Laplacian of G. In this note it is shown that if k>1 and q(G)>=n+2k-2,
then G contains cycles of length l whenever 2<l<2k+3. This bound is
asymptotically tight. It implies an asymptotic solution to a recent conjecture
about the maximum q(G) of a graph G with no cycle of a specified length.Comment: 10 pages. Version 2 takes care of some mistakes in version
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