53,923 research outputs found
A Dirac type result on Hamilton cycles in oriented graphs
We show that for each \alpha>0 every sufficiently large oriented graph G with
\delta^+(G),\delta^-(G)\ge 3|G|/8+ \alpha |G| contains a Hamilton cycle. This
gives an approximate solution to a problem of Thomassen. In fact, we prove the
stronger result that G is still Hamiltonian if
\delta(G)+\delta^+(G)+\delta^-(G)\geq 3|G|/2 + \alpha |G|. Up to the term
\alpha |G| this confirms a conjecture of H\"aggkvist. We also prove an Ore-type
theorem for oriented graphs.Comment: Added an Ore-type resul
On the swap-distances of different realizations of a graphical degree sequence
One of the first graph theoretical problems which got serious attention
(already in the fifties of the last century) was to decide whether a given
integer sequence is equal to the degree sequence of a simple graph (or it is
{\em graphical} for short). One method to solve this problem is the greedy
algorithm of Havel and Hakimi, which is based on the {\em swap} operation.
Another, closely related question is to find a sequence of swap operations to
transform one graphical realization into another one of the same degree
sequence. This latter problem got particular emphases in connection of fast
mixing Markov chain approaches to sample uniformly all possible realizations of
a given degree sequence. (This becomes a matter of interest in connection of --
among others -- the study of large social networks.) Earlier there were only
crude upper bounds on the shortest possible length of such swap sequences
between two realizations. In this paper we develop formulae (Gallai-type
identities) for these {\em swap-distance}s of any two realizations of simple
undirected or directed degree sequences. These identities improves considerably
the known upper bounds on the swap-distances.Comment: to be publishe
Distance colouring without one cycle length
We consider distance colourings in graphs of maximum degree at most and
how excluding one fixed cycle length affects the number of colours
required as . For vertex-colouring and , if any two
distinct vertices connected by a path of at most edges are required to be
coloured differently, then a reduction by a logarithmic (in ) factor against
the trivial bound can be obtained by excluding an odd cycle length
if is odd or by excluding an even cycle length . For edge-colouring and , if any two distinct edges connected by
a path of fewer than edges are required to be coloured differently, then
excluding an even cycle length is sufficient for a logarithmic
factor reduction. For , neither of the above statements are possible
for other parity combinations of and . These results can be
considered extensions of results due to Johansson (1996) and Mahdian (2000),
and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang
(2014).Comment: 14 pages, 1 figur
Cyclic cycle systems of the complete multipartite graph
In this paper, we study the existence problem for cyclic -cycle
decompositions of the graph , the complete multipartite graph with
parts of size , and give necessary and sufficient conditions for their
existence in the case that
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