A Traceability Conjecture for Oriented Graphs
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Published:2008-12-09
Issue:1
Volume:15
Page:
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ISSN:1077-8926
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Container-title:The Electronic Journal of Combinatorics
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language:
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Short-container-title:Electron. J. Combin.
Author:
Frick Marietjie,Van Aardt Susan A,Dunbar Jean E,Nielsen Morten H,Oellermann Ortrud R
Abstract
A (di)graph $G$ of order $n$ is $k$-traceable (for some $k$, $1\leq k\leq n$) if every induced sub(di)graph of $G$ of order $k$ is traceable. It follows from Dirac's degree condition for hamiltonicity that for $k\geq2$ every $k$-traceable graph of order at least $2k-1$ is hamiltonian. The same is true for strong oriented graphs when $k=2,3,4,$ but not when $k\geq5$. However, we conjecture that for $k\geq2$ every $k$-traceable oriented graph of order at least $2k-1$ is traceable. The truth of this conjecture would imply the truth of an important special case of the Path Partition Conjecture for Oriented Graphs. In this paper we show the conjecture is true for $k \leq 5$ and for certain classes of graphs. In addition we show that every strong $k$-traceable oriented graph of order at least $6k-20$ is traceable. We also characterize those graphs for which all walkable orientations are $k$-traceable.
Publisher
The Electronic Journal of Combinatorics
Subject
Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics
Cited by
5 articles.
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