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 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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01.03.2023
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| ISSN: | 0364-9024, 1097-0118 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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 of a semicomplete digraph and an arc such that has a spanning eulerian subdigraph containing . In particular, we show that if is 2‐arc‐strong, then every arc is contained in a spanning eulerian subdigraph. We then characterize the pairs of a semicomplete digraph and an arc such that has a spanning eulerian subdigraph avoiding . 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 such that every ‐arc‐strong semicomplete digraph contains a spanning eulerian subdigraph avoiding any prescribed set of arcs. We conjecture that and establish this conjecture for and when the arcs that we delete form a forest of stars. A digraph is eulerian‐connected if for any two distinct vertices , the digraph has a spanning ‐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. 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. |
| Author | Havet, Frédéric Yeo, Anders Bang‐Jensen, Jørgen |
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| Cites_doi | 10.1016/0196-6774(92)90008-Z 10.1016/S0166-218X(97)00037-1 10.1016/0095-8956(80)90061-1 10.1007/978-3-319-71840-8 10.1002/jgt.21810 10.1016/j.disc.2009.04.019 10.1016/j.amc.2020.125595 10.1007/978-1-84800-998-1 10.1007/BF01788546 10.1090/psapm/010/0114759 10.1016/j.disc.2020.112129 10.1017/S0963548397003027 |
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| References | 2015; 79 1980; 28 1987; 3 2020; 343 1992; 13 1960 1934; 7 1997; 6 2021; 389 2010; 310 1997; 79 1959; 249 e_1_2_9_11_1 e_1_2_9_10_1 e_1_2_9_13_1 e_1_2_9_12_1 e_1_2_9_8_1 e_1_2_9_7_1 e_1_2_9_6_1 e_1_2_9_5_1 e_1_2_9_4_1 e_1_2_9_3_1 Camion P. (e_1_2_9_9_1) 1959; 249 e_1_2_9_2_1 e_1_2_9_15_1 Rédei L. (e_1_2_9_14_1) 1934; 7 |
| References_xml | – volume: 343 issue: 12 year: 2020 article-title: Spanning eulerian subdigraphs avoiding k prescribed arcs in tournaments publication-title: Discrete Mathematics – volume: 389 year: 2021 article-title: Trail‐connected tournaments publication-title: Appl. Math. Computat – volume: 7 start-page: 39 year: 1934 end-page: 43 article-title: Ein kombinatorischer Satz publication-title: Acta. Litt. Sci. Szeged – volume: 6 start-page: 255 issue: 3 year: 1997 end-page: 261 article-title: Hamiltonian cycles avoiding prescribed arcs in tournaments publication-title: Combin. Prob. Comput – volume: 3 start-page: 239 issue: 3 year: 1987 end-page: 250 article-title: Hamiltonian dicycles avoiding prescribed arcs in tournaments publication-title: Graphs Combin – volume: 249 start-page: 2151 year: 1959 end-page: 2152 article-title: Chemins et circuits hamiltoniens des graphes complets publication-title: C. R. Acad. Sci. Paris – volume: 79 start-page: 119 issue: 1‐3 year: 1997 end-page: 125 article-title: Spanning local tournaments in locally semicomplete digraphs publication-title: Discrete Appl. Math – volume: 310 start-page: 1424 year: 2010 end-page: 1428 article-title: Spanning 2‐strong tournaments in 3‐strong semicomplete digraphs publication-title: Discrete Math – volume: 13 start-page: 114 issue: 1 year: 1992 end-page: 127 article-title: A polynomial algorithm for Hamiltonian‐connectedness in semicomplete digraphs publication-title: J. Algor – start-page: 113 year: 1960 end-page: 128 – volume: 79 start-page: 8 issue: 1 year: 2015 end-page: 20 article-title: Sufficient conditions for a digraph to be supereulerian publication-title: J. Graph Theory – volume: 28 start-page: 142 issue: 2 year: 1980 end-page: 163 article-title: Hamiltonian‐connected tournaments publication-title: J. Combin. Theory Ser. B – ident: e_1_2_9_8_1 doi: 10.1016/0196-6774(92)90008-Z – ident: e_1_2_9_11_1 doi: 10.1016/S0166-218X(97)00037-1 – ident: e_1_2_9_15_1 doi: 10.1016/0095-8956(80)90061-1 – ident: e_1_2_9_5_1 doi: 10.1007/978-3-319-71840-8 – ident: e_1_2_9_2_1 doi: 10.1002/jgt.21810 – ident: e_1_2_9_7_1 doi: 10.1016/j.disc.2009.04.019 – ident: e_1_2_9_13_1 doi: 10.1016/j.amc.2020.125595 – ident: e_1_2_9_4_1 doi: 10.1007/978-1-84800-998-1 – volume: 249 start-page: 2151 year: 1959 ident: e_1_2_9_9_1 article-title: Chemins et circuits hamiltoniens des graphes complets publication-title: C. R. Acad. Sci. Paris – ident: e_1_2_9_10_1 doi: 10.1007/BF01788546 – ident: e_1_2_9_12_1 doi: 10.1090/psapm/010/0114759 – ident: e_1_2_9_3_1 doi: 10.1016/j.disc.2020.112129 – volume: 7 start-page: 39 year: 1934 ident: e_1_2_9_14_1 article-title: Ein kombinatorischer Satz publication-title: Acta. Litt. Sci. Szeged – ident: e_1_2_9_6_1 doi: 10.1017/S0963548397003027 |
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| Snippet | 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... |
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| SubjectTerms | Apexes arc‐connectivity eulerian subdigraph Graph theory polynomial algorithm semicomplete digraph tournament |
| Title | Spanning eulerian subdigraphs in semicomplete digraphs |
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