7,309 research outputs found
Long properly colored cycles in edge colored complete graphs
Let denote a complete graph on vertices whose edges are
colored in an arbitrary way. Let denote the
maximum number of edges of the same color incident with a vertex of
. A properly colored cycle (path) in is a cycle (path)
in which adjacent edges have distinct colors. B. Bollob\'{a}s and P. Erd\"{o}s
(1976) proposed the following conjecture: if , then contains a properly
colored Hamiltonian cycle. Li, Wang and Zhou proved that if
, then
contains a properly colored cycle of length at least . In this paper, we improve the bound to .Comment: 8 page
Proper Hamiltonian Cycles in Edge-Colored Multigraphs
A -edge-colored multigraph has each edge colored with one of the
available colors and no two parallel edges have the same color. A proper
Hamiltonian cycle is a cycle containing all the vertices of the multigraph such
that no two adjacent edges have the same color. In this work we establish
sufficient conditions for a multigraph to have a proper Hamiltonian cycle,
depending on several parameters such as the number of edges and the rainbow
degree.Comment: 13 page
Ramsey numbers of ordered graphs
An ordered graph is a pair where is a graph and
is a total ordering of its vertices. The ordered Ramsey number
is the minimum number such that every ordered
complete graph with vertices and with edges colored by two colors contains
a monochromatic copy of .
In contrast with the case of unordered graphs, we show that there are
arbitrarily large ordered matchings on vertices for which
is superpolynomial in . This implies that
ordered Ramsey numbers of the same graph can grow superpolynomially in the size
of the graph in one ordering and remain linear in another ordering.
We also prove that the ordered Ramsey number is
polynomial in the number of vertices of if the bandwidth of
is constant or if is an ordered graph of constant
degeneracy and constant interval chromatic number. The first result gives a
positive answer to a question of Conlon, Fox, Lee, and Sudakov.
For a few special classes of ordered paths, stars or matchings, we give
asymptotically tight bounds on their ordered Ramsey numbers. For so-called
monotone cycles we compute their ordered Ramsey numbers exactly. This result
implies exact formulas for geometric Ramsey numbers of cycles introduced by
K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of
Combinatoric
A 7/9 - Approximation Algorithm for the Maximum Traveling Salesman Problem
We give a 7/9 - Approximation Algorithm for the Maximum Traveling Salesman
Problem.Comment: 6 figure
Maximum -edge-colorable subgraphs of class II graphs
A graph is class II, if its chromatic index is at least . Let
be a maximum -edge-colorable subgraph of . The paper proves best
possible lower bounds for , and structural properties of
maximum -edge-colorable subgraphs. It is shown that every set of
vertex-disjoint cycles of a class II graph with can be extended
to a maximum -edge-colorable subgraph. Simple graphs have a maximum
-edge-colorable subgraph such that the complement is a matching.
Furthermore, a maximum -edge-colorable subgraph of a simple graph is
always class I.Comment: 13 pages, 2 figures, the proof of the Lemma 1 is correcte
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
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