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Given a graph G and a function h from E ( G ) to N , the h -subdivision of G , denoted by G ( h ) , is the graph obtained from G by replacing each edge u v in G with a path P : u x u v 1 x u v 2 … x u v n − 1 v , where n = h ( u v ) . When h ( e ) = c is a constant for all e ∈ E ( G ) , we use G ( c ) to replace G ( h ) . Given a graph G , an L ( 2 , 1 ) -labeling of G is a function f from the vertex set V ( G ) to the set of all nonnegative integers such that | f ( x ) − f ( y ) | ≥ 2 if d G ( x , y ) = 1 , and | f ( x ) − f ( y ) | ≥ 1 if d G ( x , y ) = 2 . A k - L ( 2 , 1 ) -labeling is an L ( 2 , 1 ) -labeling such that no label is greater than k . The L ( 2 , 1 ) -labeling number of G , denoted by λ ( G ) , is the smallest number k such that G has a k - L ( 2 , 1 ) -labeling. We study the L ( 2 , 1 ) -labeling numbers of subdivisions of graphs in this paper. We prove that λ ( G ( 3 ) ) = Δ ( G ) + 1 for any graph G with Δ ( G ) ≥ 4 , and show that λ ( G ( h ) ) = Δ ( G ) + 1 if Δ ( G ) ≥ 5 and h is a function from E ( G ) to N so that h ( e ) ≥ 3 for all e ∈ E ( G ) , or if Δ ( G ) ≥ 4 and h is a function from E ( G ) to N so that h ( e ) ≥ 4 for all e ∈ E ( G ) . [ABSTRACT FROM AUTHOR]

Given a graph G and positive integers p, q with p≥ q, the ( p, q)-total number $\lambda_{p,q}^{T}(G)$ of G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that the labels of any two adjacent vertices are at least q apart, the labels of any two adjacent edges are at least q apart, and the difference between the labels of a vertex and its incident edges is at least p. Havet and Yu (Discrete Math 308:496-513, ) first introduced this problem and determined the exact value of $\lambda_{p,1}^{T}(K_{n})$ except for even n with p+5≤ n≤6 p−10 p+4. Their proof for showing that $\lambda _{p,1}^{T}(K_{n})\leq n+2p-3$ for odd n has some mistakes. In this paper, we prove that if n is odd, then $\lambda_{p}^{T}(K_{n})\leq n+2p-3$ if p=2, p=3, or $4\lfloor\frac{p}{2}\rfloor+3\leq n\leq4p-1$. And we extend some results that were given in Havet and Yu (Discrete Math 308:496-513, ). Beside these, we give a lower bound for $\lambda_{p,q}^{T}(K_{n})$ under the condition that q< p<2 q. [ABSTRACT FROM AUTHOR]

Abstract: Given a graph , a defensive alliance of is a set of vertices satisfying the condition that for each , at least half of the vertices in the closed neighborhood of are in . A defensive alliance is called global if every vertex in is adjacent to at least one member of the defensive alliance . The global defensive alliance number of , denoted , is the minimum size around all the global defensive alliances of . In this paper, we present an efficient algorithm to determine the global defensive alliance numbers of trees, and further give formulas to decide the global defensive alliance numbers of complete -ary trees for . We also establish upper bounds and lower bounds for and , and show that the bounds are sharp for certain . [Copyright &y& Elsevier]

Abstract: Given a graph and two positive integers with an -labeling of is a function from the vertex set to the set of all nonnegative integers such that if and if . A -labeling is an -labeling such that no label is greater than . The -labeling number of , denoted by is the smallest number such that has a -labeling. When considering the digraph , we use in place of . We study the -labeling number of a digraph in this paper. We find some relations between the -labeling number of a graph and an orientation of , and give some results for the -labeling numbers of -partite digraphs. We also study the -labeling numbers for those graphs for which the underlying graphs are paths, cycles or trees. [Copyright &y& Elsevier]

Abstract: Given a graph and a vertex subset of , the broadcasting time with respect to , denoted by , is the minimum broadcasting time when using as the broadcasting set. And the -broadcasting number, denoted by , is defined by . Given a graph and two vertex subsets , of , define , , , and for all . For all , , the -radius of , denoted by , is defined as , . In this paper, we study the relation between the -radius and the -broadcasting numbers of graphs. We also give the -radius and the -broadcasting numbers of the grid graphs, and the -broadcasting numbers of the complete -partite graphs and the hypercubes. [Copyright &y& Elsevier]