Publications (2)0 Total impact
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ABSTRACT: Let $G$ be an edge-colored graph. The color degree of a vertex $v$ of $G$, is
defined as the number of colors of the edges incident to $v$. The color number
of $G$ is defined as the number of colors of the edges in $G$. A
heterochromatic triangle is one in which every pair of edges have different
colors. In this paper we give some sufficient conditions for the existence of
heterochromatic triangles in edge-colored graphs in terms of color degree,
color number and edge number. As a corollary, a conjecture proposed by Li and
Wang (Color degree and heterochromatic cycles in edge-colored graphs, European
J. Combin. 33 (2012) 1958-1964) is confirmed.
12/2012;
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ABSTRACT: Computer or communication networks are so designed that they do not easily
get disrupted under external attack and, moreover, these are easily
reconstructible if they do get disrupted. These desirable properties of
networks can be measured by various graph parameters, such as connectivity,
toughness, scattering number, integrity, tenacity, rupture degree and
edge-analogues of some of them. Among these parameters, the tenacity and
rupture degree are two better ones to measure the stability of a network. In
this paper we consider two extremal problems on the tenacity of graphs:
Determine the minimum and maximum tenacity of graphs with given order and size.
We give a complete solution to the first problem, while for the second one, it
turns out that the problem is much more complicated than that of the minimum
case. We determine the maximum tenacity of trees and unicyclic graphs with
given order and show the corresponding extremal graphs. These results are
helpful in constructing stable networks with lower costs. The paper concludes
with a discussion of a related problem on the edge vulnerability parameters of
graphs.
09/2011;
