2,506 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
A revisit to Bang-Jensen-Gutin conjecture and Yeo's theorem
A path (cycle) is properly-colored if consecutive edges are of distinct
colors. In 1997, Bang-Jensen and Gutin conjectured a necessary and sufficient
condition for the existence of a Hamilton path in an edge-colored complete
graph. This conjecture, confirmed by Feng, Giesen, Guo, Gutin, Jensen and
Rafley in 2006, was laterly playing an important role in Lo's asymptotical
proof of Bollob\'as-Erd\H{o}s' conjecture on properly-colored Hamilton cycles.
In 1997, Yeo obtained a structural characterization of edge-colored graphs that
containing no properly colored cycles. This result is a fundamental tool in the
study of edge-colored graphs. In this paper, we first give a much shorter proof
of the Bang-Jensen-Gutin Conjecture by two novel absorbing lemmas. We also
prove a new sufficient condition for the existence of a properly-colored cycle
and then deduce Yeo's theorem from this result and a closure concept in
edge-colored graphs.Comment: 13 pages, 5 figure
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
Minimum degree conditions for monochromatic cycle partitioning
A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any
-edge-coloured complete graph has a partition into
monochromatic cycles. Here we determine the minimum degree threshold for this
property. More precisely, we show that there exists a constant such that
any -edge-coloured graph on vertices with minimum degree at least has a partition into monochromatic cycles. We also
provide constructions showing that the minimum degree condition and the number
of cycles are essentially tight.Comment: 22 pages (26 including appendix
- …