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Balanced Subeulerian Signed Graphs and Signed Line Graphs
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A signed graph S=(Su,a)has an underlying graph Su and 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 Su is 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'.
signed graphsigned line graphbalanced eulerianbalanced subeulerian
Juan LIU、Hong YANG、Xindong ZHANG、Hongjian LAI
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College of Big Data Statistics,Guizhou University of Finance and Economics,Guizhou 550025,P.R.China
College of Mathematics and System Sciences,Xinjiang University,Xinjiang 830046,P.R.China
Department of Mathematics,West Virginia University,Morgantown 26506,USA
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