# Search results for: Xueliang Li

Bulletin of the Malaysian Mathematical Sciences Society > 2019 > 42 > 1 > 381-390

*P*in a total-colored graph

*G*is called a total-proper path if (1) any two adjacent edges of

*P*are assigned distinct colors; (2) any two adjacent internal vertices of

*P*are assigned distinct colors; and (3) any internal vertex of

*P*is assigned a distinct color from its incident edges of

*P*. The total-colored...

Bulletin of the Malaysian Mathematical Sciences Society > 2018 > 41 > 4 > 1681-1695

*T*in an edge-colored graph is a

*proper tree*if no two adjacent edges of

*T*receive the same color. Let

*G*be a connected graph of order

*n*and

*k*be a fixed integer with $$2\le k\le n$$ 2≤k≤n . For a vertex subset $$S \subseteq V(G)$$ S⊆V(G) with $$\left| S\right| \ge 2$$ S≥2 , a tree containing all the vertices of

*S*in

*G*is called an

*S*-tree. An edge-coloring of

*G*is called a

*k*-

*proper coloring*...

Bulletin of the Malaysian Mathematical Sciences Society > 2018 > 41 > 3 > 1199-1209

*P*in an edge-colored graph

*G*is called a proper path if no two adjacent edges of

*P*are colored the same, and

*G*is proper connected if every two vertices of

*G*are connected by a proper path in

*G*. The proper connection number of a connected graph

*G*, denoted by $$\textit{pc}(G)$$ pc(G) , is the minimum number of colors that are needed to make

*G*proper connected. In this paper, we investigate the...

Bulletin of the Malaysian Mathematical Sciences Society > 2017 > 40 > 1 > 321-333

*G*. Let $$S(G^\sigma )$$ S ( G σ ) be the skew-adjacency matrix of $$G^\sigma $$ G σ . The skew energy of $$G^\sigma $$ G σ is defined as the sum of the absolute values of all eigenvalues of $$S(G^\sigma...

Bulletin of the Malaysian Mathematical Sciences Society > 2016 > 39 > 3 > 1225-1236

*G*is said to be a rainbow tree if no two edges on the tree share the same color. Given two positive integers

*k*, $$\ell $$ ℓ with $$k\ge 3$$ k ≥ 3 , the $$(k,\ell )$$ ( k , ℓ ) -

*rainbow index*$$rx_{k,\ell }(G)$$ r x k , ℓ ( G ) of

*G*is the minimum number of colors needed in an edge-coloring of

*G*such that for any set

*S*of...

Bulletin of the Malaysian Mathematical Sciences Society > 2016 > 39 > 2 > 765-771

*G*is said to be a rainbow tree if no two edges on the tree share the same color. Given two positive integers

*k*, $$\ell $$ ℓ with $$k\ge 3$$ k ≥ 3 , the $$(k,\ell )$$ ( k , ℓ )

*-rainbow index*$$rx_{k,\ell }(G)$$ r x k , ℓ ( G ) of

*G*is the minimum number of colors needed in an edge-coloring of

*G*such that for any set

*S*of...

Bulletin of the Malaysian Mathematical Sciences Society > 2016 > 39 > 1 > 409-425

*G*be an edge-colored graph. A rainbow (heterochromatic, or multicolored) path of

*G*is such a path in which no two edges have the same color. Let the color degree of a vertex

*v*to be the number of different colors that are used on edges incident to

*v*, and denote it by $$d^c(v)$$ d c ( v ) . In a previous paper, we showed that if $$d^c(v)\ge k$$ d c ( v ) ≥ k (color...

Bulletin of the Malaysian Mathematical Sciences Society > 2017 > 40 > 4 > 1769-1779

*monochromatically-connecting coloring*(MC-coloring, for short) if there is a monochromatic path joining any two vertices, which was introduced by Caro and Yuster. Let

*mc*(

*G*) denote the maximum number of colors used in an MC-coloring of a graph

*G*. Note that an MC-coloring does not exist if

*G*is not connected, in which case we simply let $$mc(G)=0$$ mc(G)=0...

Bulletin of the Malaysian Mathematical Sciences Society > 2018 > 41 > 1 > 415-425

*vertex-proper path*if any two internal adjacent vertices differ in color. A vertex-colored graph is

*proper vertex*

*k*-

*connected*if any two vertices of the graph are connected by

*k*disjoint vertex-proper paths of the graph. For a

*k*-connected graph

*G*, the

*proper vertex*

*k*-

*connection number*of

*G*, denoted by $$pvc_{k}(G)$$ p v c k ( G ) , is defined as the...

Bulletin of the Malaysian Mathematical Sciences Society > 2015 > 38 > 3 > 1235-1241

Bulletin of the Malaysian Mathematical Sciences Society > 2015 > 38 > 4 > 1627-1635