Spanning eulerian subdigraphs in semicomplete digraphs
A digraph is eulerian if it is connected and every vertex has its in‐degree equal to its out‐degree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. In this paper, we first characterize the pairs (D,a) $(D,a)$ of a semicomplete digraph D $D$ and an arc a $a$...
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| Vydáno v: | Journal of graph theory Ročník 102; číslo 3; s. 578 - 606 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Hoboken
Wiley Subscription Services, Inc
01.03.2023
|
| Témata: | |
| ISSN: | 0364-9024, 1097-0118 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | A digraph is eulerian if it is connected and every vertex has its in‐degree equal to its out‐degree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. In this paper, we first characterize the pairs (D,a) $(D,a)$ of a semicomplete digraph D $D$ and an arc a $a$ such that D $D$ has a spanning eulerian subdigraph containing a $a$. In particular, we show that if D $D$ is 2‐arc‐strong, then every arc is contained in a spanning eulerian subdigraph. We then characterize the pairs (
D
,
a
) $(D,a)$ of a semicomplete digraph D $D$ and an arc a $a$ such that D $D$ has a spanning eulerian subdigraph avoiding a $a$. In particular, we prove that every 2‐arc‐strong semicomplete digraph has a spanning eulerian subdigraph avoiding any prescribed arc. We also prove the existence of a (minimum) function f
(
k
) $f(k)$ such that every f
(
k
) $f(k)$‐arc‐strong semicomplete digraph contains a spanning eulerian subdigraph avoiding any prescribed set of k $k$ arcs. We conjecture that f
(
k
)
=
k
+
1 $f(k)=k+1$ and establish this conjecture for k
≤
3 $k\le 3$ and when the k $k$ arcs that we delete form a forest of stars. A digraph D $D$ is eulerian‐connected if for any two distinct vertices x
,
y $x,y$, the digraph D $D$ has a spanning (
x
,
y
) $(x,y)$‐trail. We prove that every 2‐arc‐strong semicomplete digraph is eulerian‐connected. All our results may be seen as arc analogues of well‐known results on hamiltonian paths and cycles in semicomplete digraphs. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0364-9024 1097-0118 |
| DOI: | 10.1002/jgt.22888 |