Spanning Eulerian Subdigraphs in Jump Digraphs 
Received:August 19, 2021 Revised:February 19, 2022 
Key Words:
supereulerian digraph line digraph jump digraph weakly trailconnected strongly trailconnected

Fund Project: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) and the Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant No.2022D01E13). 
Author Name  Affiliation  Juan LIU  College of Big Data Statistics, Guizhou University of Finance and Economics, Guizhou 550025, P. R. China  Hong YANG  College of Mathematics and System Sciences, Xinjiang University, Xinjiang 830046, P. R. China  Hongjian LAI  Department of Mathematics, West Virginia University, Morgantown 26506, USA  Xindong ZHANG  School of Mathematical Sciences, Xinjiang Normal University, Xinjiang 830017, P. R. China 

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Abstract: 
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 $u_{1}, v_{1}, u_{2},v_{2}$ in $D$ such that $a=(u_{1},v_{1}),b=(u_{2},v_{2})$ and $v_{1}\not=u_{2}$. In this paper, we give a well characterized directed multigraph families $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$, and prove that a jump digraph $J(D)$ of a directed multigraph $D$ is strongly connected if and only if $D\not\in \mathcal{H}_{1}$. Specially, $J(D)$ is weakly connected if and only if $D\not\in \mathcal{H}_{2}$. The following results are obtained: (i) There exists a family $\mathcal{D}$ of wellcharacterized directed multigraphs such that strongly connected jump digraph $J(D)$ of directed multigraph is strongly trailconnected if and only if $D\not\in \mathcal{D}$. (ii) Every strongly connected jump digraph $J(D)$ of directed multigraph $D$ is weakly trailconnected, and so is supereulerian. (iii) Every weakly connected jump digraph $J(D)$ of directed multigraph $D$ has a spanning trail. 
Citation: 
DOI:10.3770/j.issn:20952651.2022.05.001 
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