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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]

Abstract: All-to-all communication occurs in many important applications in parallel processing. In this paper, we study the all-to-all broadcast number (the shortest time needed to complete the all-to-all broadcast) of graphs under the assumption that: each vertex can use all of its links at the same time, and each communication link is half duplex and can carry only one message at a unit of time. We give upper and lower bounds for the all-to-all broadcast number of graphs and give formulas for the all-to-all broadcast number of trees, complete bipartite graphs and double loop networks under this model. [Copyright &y& Elsevier]