13 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
General Models for Variations of the Even Cycle Problem
We consider three related problems in grap
Color degree and alternating cycles in edge-colored graphs
AbstractLet G be an edge-colored graph. An alternating cycle of G is a cycle of G in which any two consecutive edges have distinct colors. Let dc(v), the color degree of a vertex v, be defined as the maximum number of edges incident with v that have distinct colors. In this paper, we study color degree conditions for the existence of alternating cycles of prescribed length
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
Note on alternating directed cycles
The problem of the existence of an alternating simple dicycle in a 2-arc-coloured digraph is considered. This is a generalization of the alternating cycle problem in 2-edge-coloured graphs and the even dicycle problem (both are polynomial time solvable). We prove that the alternating dicycle problem is NP -complete. Let f(n)(g(n), resp.) be the minimum integer such that if every monochromatic indegree and outdegree in a strongly connected 2-arc-coloured digraph (any 2-arc-coloured digraph, resp.) D is at least f(n)(g(n), resp.), then D has an alternating simple dicycle. We show that f(n) = #(log n) and g(n) = #(log n). ? 1998 Elsevier Science B.V. All rights reserved Keywords: Alternating cycles; Even cycles; Edge-coloured directed graph 1. Introduction, terminology and notation We shall assume that the reader is familiar with the standard terminology on graphs and digraphs and refer the reader to [4]. We consider digraphs without loops and multiple arcs. The arcs of digraphs are colo..