Balanced Subeulerian Signed Graphs and Signed Line Graphs

A signed graph S =(Su, σ) has an underlying graph Suand a function σ : E(Su)-→{+,-}. Let E-(S) denote the set of negative edges of S. Then S is eulerian signed graph(or subeulerian signed graph, or balanced eulerian signed graph, respectively) if Suis eulerian(or subeulerian, or eulerian and |E-(S)| is even, respectively). We say that S is balanced subeulerian signed graph if there exists a balanced eulerian signed graph S′ such that S′ is spanned by S.The signed line graph L(S) of a signed graph S is a signed graph with the vertices of L(S) being the edges of S, where an edge eiej is in L(S) if and only if the edges ei and ej of S have a vertex in common in S such that an edge eiej in L(S) is negative if and only if both edges ei and ej are negative in S. In this paper, two families of signed graphs S and S ′ are identified, which are applied to characterize balanced subeulerian signed graphs and balanced subeulerian signed line graphs. In particular, it is proved that a signed graph S is balanced subeulerian if and only if S?S, and that a signed line graph of signed graph S is balanced subeulerian if and only if S?S ′.