On forcibly -connected and forcibly -arc-connected digraphic sequences☆
Section snippets
Introduction and preliminaries
In this paper, we work on the study of digraphic sequences. For a with and and an arc , we say that is the and is the of the arc . Here, we study simple digraphs which have neither multiple arcs (arcs with the same head and the same tail) nor loops. For , the - of is and the - of is . Their cardinalities are the - of ,
-connected digraphs
In this section, let be a digraphic sequence, we will develop the theory of forcibly -connected digraphic sequences for . A digraph is -connected (or -arc-connected) is equivalent to saying that it is strong.
Theorem 2 If a digraphic sequence satisfies each of the following for , then is forcibly strong: , or , or .
Proof Under the conditions stated in Theorem 2, we conversely suppose that
-arc-connected digraphs
In this section, let be integers and let and be sequences of nonnegative integers, where . Define for each integer and each integer . The notations mentioned above will be used throughout this section. Also we use the same notation for as in the preceding section. Then we obtain the following conclusions which are of
Acknowledgments
The authors wish to acknowledge anonymous reviewers for their valuable comments.
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The research is supported by NSFXJ (2020D04046), NSFC, China (11861066), NSFXJ (2021D01C116).