5,305 research outputs found
Odd properly colored cycles in edge-colored graphs
It is well-known that an undirected graph has no odd cycle if and only if it
is bipartite. A less obvious, but similar result holds for directed graphs: a
strongly connected digraph has no odd cycle if and only if it is bipartite. Can
this result be further generalized to more general graphs such as edge-colored
graphs? In this paper, we study this problem and show how to decide if there
exists an odd properly colored cycle in a given edge-colored graph. As a
by-product, we show how to detect if there is a perfect matching in a graph
with even (or odd) number of edges in a given edge set
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
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
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
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