543 research outputs found
New bounds on even cycle creating Hamiltonian paths using expander graphs
We say that two graphs on the same vertex set are -creating if their union
(the union of their edges) contains as a subgraph. Let be the
maximum number of pairwise -creating Hamiltonian paths of . Cohen,
Fachini and K\"orner proved In this paper we close the superexponential gap
between their lower and upper bounds by proving
We also improve the
previously established upper bounds on for , and we present
a small improvement on the lower bound of F\"uredi, Kantor, Monti and Sinaimeri
on the maximum number of so-called pairwise reversing permutations. One of our
main tools is a theorem of Krivelevich, which roughly states that (certain
kinds of) good expanders contain many Hamiltonian paths.Comment: 14 pages, LaTeX2e; v2: updated Footnote 1 on Page 5; v3: revised
version incorporating suggestions by the referees (the changes are mainly in
Section 5); v4: final version to appear in Combinatoric
New bounds on even cycle creating Hamiltonian paths using expander graphs
We say that two graphs on the same vertex set are -creating if their union
(the union of their edges) contains as a subgraph. Let be the
maximum number of pairwise -creating Hamiltonian paths of . Cohen,
Fachini and K\"orner proved In this paper we close the superexponential gap
between their lower and upper bounds by proving
We also improve the
previously established upper bounds on for , and we present
a small improvement on the lower bound of F\"uredi, Kantor, Monti and Sinaimeri
on the maximum number of so-called pairwise reversing permutations. One of our
main tools is a theorem of Krivelevich, which roughly states that (certain
kinds of) good expanders contain many Hamiltonian paths.Comment: 14 pages, LaTeX2e; v2: updated Footnote 1 on Page 5; v3: revised
version incorporating suggestions by the referees (the changes are mainly in
Section 5); v4: final version to appear in Combinatoric
On hamiltonian colorings of block graphs
A hamiltonian coloring c of a graph G of order p is an assignment of colors
to the vertices of G such that for every two
distinct vertices u and v of G, where D(u,v) denoted the detour distance
between u and v. The value hc(c) of a hamiltonian coloring c is the maximum
color assigned to a vertex of G. The hamiltonian chromatic number, denoted by
hc(G), is the min{hc(c)} taken over all hamiltonian coloring c of G. In this
paper, we present a lower bound for the hamiltonian chromatic number of block
graphs and give a sufficient condition to achieve the lower bound. We
characterize symmetric block graphs achieving this lower bound. We present two
algorithms for optimal hamiltonian coloring of symmetric block graphs.Comment: 12 pages, 1 figure. A conference version appeared in the proceedings
of WALCOM 201
Recent trends and future directions in vertex-transitive graphs
A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade
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