Signed degree sequences in signed multipartite graphs

Abstract

A signed  k -partite graph (signed multipartite graph) is a  k -partite graph in which each edge is assigned a positive or a negative sign. If  G ( V 1 , V 2 ,  · · ·  , V k )  is a signed  k -partite graph with  V i  = { v i 1 ,v i 2 , ···  ,v in i } ,  1  ≤  i  ≤  k , the signed degree of  v ij  is  sdeg ( v ij ) =  d ij  =  d + ij −  d − ij , where  1 ≤ i ≤ k , 1 ≤ j ≤ n i  and  d + ij ( d − ij ) is the number of positive (negative) edges incident with  v ij . The sequences  α i  = [ d i 1 ,d i 2 , ···  ,d in i ] ,  1  ≤  i  ≤  k , are called the signed degree sequences of  G ( V 1 ,V 2 , ···  ,V k ) . The set of distinct signed degrees of the vertices in a signed  k -partite graph  G ( V 1 , V 2 ,  · · ·  , V k )  is called its signed degree set. In this paper, we characterize signed degree sequences of signed  k -partite graphs. Also, we give the existence of signed  k -partite graphs with given signed degree sets.