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Meng, Wei, Grueter, Steffen, Guo, Yubao, Kapolke, Manu, Meesker, Simon. (1398). t-Pancyclic Arcs in Tournaments. , 4(2), 123-130. doi: 10.22049/cco.2019.26333.1097
Wei Meng; Steffen Grueter; Yubao Guo; Manu Kapolke; Simon Meesker. "t-Pancyclic Arcs in Tournaments". , 4, 2, 1398, 123-130. doi: 10.22049/cco.2019.26333.1097
Meng, Wei, Grueter, Steffen, Guo, Yubao, Kapolke, Manu, Meesker, Simon. (1398). 't-Pancyclic Arcs in Tournaments', , 4(2), pp. 123-130. doi: 10.22049/cco.2019.26333.1097
Meng, Wei, Grueter, Steffen, Guo, Yubao, Kapolke, Manu, Meesker, Simon. t-Pancyclic Arcs in Tournaments. , 1398; 4(2): 123-130. doi: 10.22049/cco.2019.26333.1097

t-Pancyclic Arcs in Tournaments

Communications in Combinatorics and Optimization
مقاله 5، دوره 4، شماره 2، اسفند 2019، صفحه 123-130    اصل مقاله (354.83 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2019.26333.1097
نویسندگان
Wei Meng* 1؛ Steffen Grueter2؛ Yubao Guo2؛ Manu Kapolke2؛ Simon Meesker2
1School of Mathematical Sciences, Shanxi University, 030006 Taiyuan, China
2Lehrstuhl C fuer Mathematik, RWTH Aachen University, 52056 Aachen, Germany
چکیده
Let $T$ be a non-trivial tournament. An arc is \emph{$t$-pancyclic} in $T$, if it is contained in a cycle of length $\ell$ for every $t\leq \ell \leq |V(T)|$. Let $p^t(T)$ denote the number of $t$-pancyclic arcs in $T$ and $h^t(T)$ the maximum number of $t$-pancyclic arcs contained in the same Hamiltonian cycle of $T$. Moon ( J. Combin. Inform. System Sci., 19 (1994), 207-214) showed that $h^3(T)\geq3$ for any non-trivial strong tournament $T$ and characterized the tournaments with $h^3(T)= 3$. In this paper, we generalize Moon's theorem by showing that $h^t(T)\geq t$ for every $3\leq t\leq |V(T)|$ and characterizing all tournaments which satisfy $h^t(T)= t$. We also present all tournaments which fulfill $p^t(T)= t$. 
کلیدواژه‌ها
tournament؛ pancyclicity؛ t-pancyclic arc
مراجع
[1] B. Alspach, Cycles of each length in regular tournaments, Canad. Math. Bull. 10 (1967), no. 2, 283–286.

[2] R.J. Douglas, Tournaments that admit exactly one hamiltonian circuit, Proc. London Math. Soc. 21 (1970), no. 4, 716–730.

[3] F. Havet, Pancyclic arcs and connectivity in tournaments, J. Graph Theory 47 (2004), no. 2, 87–110.

[4] J.W. Moon, On k-cyclic and pancyclic arcs in strong tournaments, J. Combin. Inform. System Sci. 19 (1994), 207–214.

[5] C. Thomassen, Hamiltonian-connected tournaments, J. Combin. Theory Ser. B 28 (1980), no. 2, 142–163.

[6] A. Yeo, The number of pancyclic arcs in ak-strong tournament, J. Graph Theory 50 (2005), no. 3, 212–219.

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