4 research outputs found
Color degree and color neighborhood union conditions for long heterochromatic paths in edge-colored graphs
Let be an edge-colored graph. A heterochromatic (rainbow, or
multicolored) path of is such a path in which no two edges have the same
color. Let denote the color degree and denote the color
neighborhood of a vertex of . In a previous paper, we showed that if
(color degree condition) for every vertex of , then
has a heterochromatic path of length at least , and
if (color neighborhood union condition) for every
pair of vertices and of , then has a heterochromatic path of
length at least . Later, in another paper we first
showed that if , has a heterochromatic path of length at least
, and then, based on this we use induction on and showed that if
, then has a heterochromatic path of length at least
. In the present paper, by using a simpler approach
we further improve the result by showing that if , has a
heterochromatic path of length at least , which
confirms a conjecture by Saito. We also improve a previous result by showing
that under the color neighborhood union condition, has a heterochromatic
path of length at least .Comment: 12 page
Rainbow C_4's and Directed C_4's: the Bipartite Case Study
In this paper we obtain a new sufficient condition for the existence of
directed cycles of length 4 in oriented bipartite graphs. As a corollary, a
conjecture of H. Li is confirmed. As an application, a sufficient condition for
the existence of rainbow cycles of length 4 in bipartite edge-colored graphs is
obtained.Comment: 9 pages, accepted by Bull. Malays. Math. Sci. So
Long rainbow path in properly edge-colored complete graphs
Let be an edge-colored graph. A rainbow (heterochromatic, or
multicolored) path of is such a path in which no two edges have the same
color. Let the color degree of a vertex be the number of different colors
that are used on the edges incident to , and denote it to be . It
was shown that if for every vertex of , then has a
rainbow path of length at least . In
the present paper, we consider the properly edge-colored complete graph
only and improve the lower bound of the length of the longest rainbow path by
showing that if , there must have a rainbow path of length no less
than .Comment: 12 page
A note on heterochromatic cycles of length 4 in edge-colored graphs
Let be an edge-colored graph. A heterochromatic cycle of is one in
which every two edges have different colors. For a vertex , let
denote the set of colors which are assigned to the edges incident to
. In this note we prove that contains a heterochromatic cycle of length
4 if has vertices and for every pair
of vertices and of . This extends a result of Broersma et al. on the
existence of heterochromatic cycles of length 3 or 4