# Search results for: Xueliang Li

Graphs and Combinatorics > 2018 > 34 > 6 > 1553-1563

*G*is

*conflict-free connected*if every two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The

*conflict-free connection number*of a connected graph

*G*, denoted by

*cfc*(

*G*), is the smallest number of colors needed in order to make

*G*conflict-free connected. For a graph

*G*, let

*C*(

*G*) be the subgraph of

*G*induced by its set of cut-edges...

Graphs and Combinatorics > 2017 > 33 > 4 > 999-1008

*G*is called

*a rainbow tree*if no two edges of it are assigned the same color. For a vertex subset $$S\subseteq V(G)$$ S ⊆ V ( G ) , a tree is called an

*S*-

*tree*if it connects

*S*in

*G*. A

*k*-

*rainbow coloring*of

*G*is an edge-coloring of

*G*having the property that for every set

*S*of

*k*vertices of

*G*, there exists a rainbow

*S*-tree in

*G*. The minimum number...

Graphs and Combinatorics > 2016 > 32 > 5 > 1829-1841

*G*, the minimum number of colors that are needed to make

*G*rainbow connected is called the rainbow connection number of

*G*, denoted by rc(

*G*). In this paper,...

Graphs and Combinatorics > 2015 > 31 > 6 > 2231-2259

*generalized local connectivity*...

Graphs and Combinatorics > 2015 > 31 > 1 > 141-147

*G*is

*rainbow connected*if every pair of vertices of

*G*are connected by a path whose edges have distinct colors. The

*rainbow connection number*

*rc*(

*G*) of

*G*is defined to be the minimum integer

*t*such that there exists an edge-coloring of

*G*with

*t*colors that makes

*G*rainbow connected. For a graph

*G*without any cut vertex, i.e., a 2-connected graph, of order

*n*, it was proved that...

Graphs and Combinatorics > 2014 > 30 > 4 > 949-955

*G*, where adjacent edges may have the same color, is

*rainbow connected*if every two vertices of

*G*are connected by a path whose edges have distinct colors. A graph

*G*is

*d-rainbow connected*if one can use

*d*colors to make

*G*rainbow connected. For integers

*n*and

*d*let

*t*(

*n, d*) denote the minimum size (number of edges) in

*d*-rainbow connected graphs of order

*n*. Schiermeyer got some...

Graphs and Combinatorics > 2013 > 29 > 5 > 1235-1247

*G*is

*rainbow connected*if every two vertices of

*G*are connected by a path whose edges have distinct colors. The

*rainbow connection number*of

*G*, denoted by

*rc*(

*G*), is the minimum number of colors that are needed to make

*G*rainbow connected. In this paper we give a Nordhaus–Gaddum-type result for the rainbow connection number. We prove that if

*G*and $${\overline{G}}$$ are both...

Graphs and Combinatorics > 2013 > 29 > 5 > 1471-1475

*G*is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph

*G*, denoted by

*rc*(

*G*), is the smallest number of colors that are needed in order to make

*G*rainbow connected. In this paper, we proved that

*rc*(

*G*) ≤ 3(

*n*+ 1)/5 for all 3-connected graphs.

Graphs and Combinatorics > 2013 > 29 > 6 > 1733-1739

*G*be a connected graph. The notion of rainbow connection number

*rc*(

*G*) of a graph

*G*was introduced by Chartrand et al. (Math Bohem 133:85–98, 2008). Basavaraju et al. (arXiv:1011.0620v1 [math.CO], 2010) proved that for every bridgeless graph

*G*with radius

*r*, $${rc(G)\leq r(r+2)}$$ and the bound is tight. In this paper, we show that for a connected graph

*G*with radius

*r*and center vertex

*u*,...

Graphs and Combinatorics > 2013 > 29 > 1 > 1-38

*k*...

Graphs and Combinatorics > 2012 > 28 > 2 > 251-263

*G*, where adjacent edges may be colored the same, is called a rainbow path if no two edges of it are colored the same. A nontrivial connected graph

*G*is rainbow connected if for any two vertices of

*G*there is a rainbow path connecting them. The rainbow connection number of

*G*, denoted

*rc*(

*G*), is defined as the smallest number of colors such that

*G*is rainbow connected...

Graphs and Combinatorics > 2008 > 24 > 4 > 237-263

Graphs and Combinatorics > 2008 > 24 > 6 > 551-562

*P*

_{4}-transformation is one-to-one on the set of graphs with minimum degree at least 3, and if graphs

*G*and

*G*

^{'}have minimum degree at least 3 then any isomorphism from the

*P*

_{4}-graph

*P*

_{4}(

*G*) to the

*P*

_{4}-graph

*P*

_{4}(

*G*

^{'}) is induced by a vertex-isomorphism from

*G*to

*G*

^{'}unless

*G*and

*G*

^{'}both belong to a special family of graphs.

Graphs and Combinatorics > 2006 > 22 > 4 > 497-502

*g*,

*f*)-factor of a graph

*G*into the problem of finding a minimum perfect matching in a weighted simple graph

*G**. Using the structure of

*G**, a necessary and sufficient condition for the existence of an even factor is derived.

Graphs and Combinatorics > 2004 > 20 > 3 > 413-422

*G*be a 4-connected graph. For an edge

*e*of

*G*, we do the following operations on

*G*: first, delete the edge

*e*from

*G*, resulting the graph

*G*−

*e*; second, for all vertices

*x*of degree 3 in

*G*−

*e*, delete

*x*from

*G*−

*e*and then completely connect the 3 neighbors of

*x*by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by

*G*⊖

*e*. If

*G*⊖

*e*is 4-connected,...

Graphs and Combinatorics > 2004 > 20 > 3 > 423-434

Graphs and Combinatorics > 2002 > 18 > 1 > 193-200