A jump digraph J(D)of a directed multigraph D has as its vertex set being A(D),the set of arcs of D;where(a,b)is an arc of J(D)if and only if there are vertices u1,v1,u2,v2in D such that a=(u1,v1),b=(u2,v2)and v1≠u2....A jump digraph J(D)of a directed multigraph D has as its vertex set being A(D),the set of arcs of D;where(a,b)is an arc of J(D)if and only if there are vertices u1,v1,u2,v2in D such that a=(u1,v1),b=(u2,v2)and v1≠u2.In this paper,we give a well characterized directed multigraph families H1and H2,and prove that a jump digraph J(D)of a directed multigraph D is strongly connected if and only if D?H1.Specially,J(D)is weakly connected if and only if D?H2.The following results are obtained:(ⅰ)There exists a family D of wellcharacterized directed multigraphs such that strongly connected jump digraph J(D)of directed multigraph is strongly trail-connected if and only if D?D.(ⅱ)Every strongly connected jump digraph J(D)of directed multigraph D is weakly trail-connected,and so is supereulerian.(ⅲ)Every weakly connected jump digraph J(D)of directed multigraph D has a spanning trail.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.11761071,11861068)Guizhou Key Laboratory of Big Data Statistical Analysis,China(Grant No.[2019]5103)the Natural Science Foundation of Xinjiang Uygur Autonomous Region(Grant No.2022D01E13)。
文摘A jump digraph J(D)of a directed multigraph D has as its vertex set being A(D),the set of arcs of D;where(a,b)is an arc of J(D)if and only if there are vertices u1,v1,u2,v2in D such that a=(u1,v1),b=(u2,v2)and v1≠u2.In this paper,we give a well characterized directed multigraph families H1and H2,and prove that a jump digraph J(D)of a directed multigraph D is strongly connected if and only if D?H1.Specially,J(D)is weakly connected if and only if D?H2.The following results are obtained:(ⅰ)There exists a family D of wellcharacterized directed multigraphs such that strongly connected jump digraph J(D)of directed multigraph is strongly trail-connected if and only if D?D.(ⅱ)Every strongly connected jump digraph J(D)of directed multigraph D is weakly trail-connected,and so is supereulerian.(ⅲ)Every weakly connected jump digraph J(D)of directed multigraph D has a spanning trail.
