A signed graph S=(S^(u),σ)has an underlying graph Suand a functionσ:E(S^(u))-→{+,-}.Let E^(-)(S)denote the set of negative edges of S.Then S is eulerian signed graph(or subeulerian signed graph,or balanced eulerian...A signed graph S=(S^(u),σ)has an underlying graph Suand a functionσ:E(S^(u))-→{+,-}.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 e_(i)and e_(j)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′.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.12261016)。
文摘A signed graph S=(S^(u),σ)has an underlying graph Suand a functionσ:E(S^(u))-→{+,-}.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 e_(i)and e_(j)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′.
