Signed (b, k)-Edge Covers in Graphs

Let be a simple graph with vertex set and edge set . Let have at least vertices of degree at least , where and b are positive integers. A function is said to be a signed -edge cover of G if G ( ) V G ( ) e E v ( ) E G G : ( f E k b k ) { 1,1} G   ( , ) b k ( ) f e b    for at least vertices of , where . The value k v G ( ) = {uv E( ( ) E v G u N v   ) | } ( ) min ( ) G e E f e   , taking over all signed -edge covers ( , ) b k f of G is called the signed -edge cover number of and denoted by ( , ) b k G , b k ( ) G  . In this paper we give some bounds on the signed -edge cover number of graphs. ( , b k)