Traceability of -traceable oriented graphs (original) (raw)

Extending Properties of Tournaments to k-Traceable Oriented Graphs (11frg171)

A graph or digraph is hamiltonian if it contains a cycle that visits every vertex, and traceable if it contains a path that visits every vertex. A (di)graph is k-traceable if each of its induced subdigraphs of order k is traceable. A digraph D is strong if for every pair u, v of vertices in D there is a directed path from u to v and a directed path from v to u.

An Iterative Approach to the Traceability Conjecture for Oriented Graphs

The Electronic Journal of Combinatorics, 2013

A digraph is kkk-traceable if its order is at least kkk and each of its subdigraphs of order kkk is traceable. The Traceability Conjecture (TC) states that for kgeq2k\geq 2kgeq2 every kkk-traceable oriented graph of order at least 2k−12k-12k−1 is traceable. It has been shown that for 2leqkleq62\leq k\leq 62leqkleq6, every kkk-traceable oriented graph is traceable. We develop an iterative procedure to extend previous results regarding the TC. In particular, we prove that every 777-traceable oriented graph of order at least 9 is traceable and every 8-traceable graph of order at least 14 is traceable.

Progress on the Traceability Conjecture for Oriented Graphs

Discrete mathematics & theoretical computer science DMTCS

A digraph is k-traceable if each of its induced subdigraphs of order k is traceable. The Traceability Conjecture is that for k≥2 every k-traceable oriented graph of order at least 2k-1 is traceable. The conjecture has been proved for k≤5. We prove that it also holds for k=6.

Cycles in -traceable oriented graphs

Discrete Mathematics, 2011

A digraph of order at least k is termed k-traceable if each of its subdigraphs of order k is traceable. It turns out that several properties of tournaments-i.e., the 2-traceable oriented graphs-extend to k-traceable oriented graphs for small values of k. For instance, the authors together with O. Oellermann have recently shown that for k = 2, 3, 4, 5, 6, all k-traceable oriented graphs are traceable. Moon [J.W. Moon, On subtournaments of a tournament, Canad. Math. Bull. 9(3) (1966) 297-301] observed that every nontrivial strong tournament T is vertex-pancyclic-i.e., through each vertex there is a cycle of every length from 3 up to the order of T. The present paper reports results pertaining to various cycle properties of strong k-traceable oriented graphs and explores the extent to which pancyclicity is retained by strong k-traceable oriented graphs. For each k ≥ 2 there are infinitely many k-traceable oriented graphs-e.g. tournaments. However, we establish an upper bound (linear in k) on the order of k-traceable oriented graphs having a strong component with girth greater than 3. As an application of our findings, we show that the Path Partition Conjecture holds for 1-deficient oriented graphs having a strong component with girth at least 6. (A digraph is 1-deficient if its order is exactly one more than the order of its longest paths.

A Linear Bound towards the Traceability Conjecture

The Electronic Journal of Combinatorics

A digraph is k-traceable if its order is at least k and each of its subdigraphs of order k is traceable. An oriented graph is a digraph without 2-cycles. The 2-traceable oriented graphs are exactly the nontrivial tournaments, so k-traceable oriented graphs may be regarded as generalized tournaments. It is well-known that all tournaments are traceable. We denote by t(k) the smallest integer bigger than or equal to k such that every k-traceable oriented graph of order at least t(k) is traceable. The Traceability Conjecture states that t(k) ≤ 2k-1 for every k ≥ 2 [van Aardt, Dunbar, Frick, Nielsen and Oellermann, A traceability conjecture for oriented graphs, Electron. J. Combin., 15(1):#R150, 2008]. We show that for k ≥ 2, every k-traceable oriented graph with independence number 2 and order at least 4k-12 is traceable. This is the last open case in giving an upper bound for t(k) that is linear in k.

A Traceability Conjecture for Oriented Graphs

The Electronic Journal of Combinatorics, 2008

A (di)graph G of order n is k-traceable (for some k, 1 k 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 2 every k-traceable graph of order at least 2k 1 is hamiltonian. The same is true for strong oriented graphs when k = 2;

Partitioning Transitive Tournaments into Isomorphic Digraphs

Order

In an earlier paper (see Sali and Simonyi Eur. J. Combin. 20, 93-99, 1999) the first two authors have shown that self-complementary graphs can always be oriented in such a way that the union of the oriented version and its isomorphically oriented complement gives a transitive tournament. We investigate the possibilities of generalizing this theorem to decompositions of the complete graph into three or more isomorphic graphs. We find that a complete characterization of when an orientation with similar properties is possible seems elusive. Nevertheless, we give sufficient conditions that generalize the earlier theorem and also imply that decompositions of odd vertex complete graphs to Hamiltonian cycles admit such an orientation. These conditions are further generalized and some necessary conditions are given as well.

Path decompositions of digraphs

Bulletin of the Australian Mathematical Society, 1974

A path decomposition of a digraph G (having no loops or multiple arcs) is a family of simple paths such that every arc of G lies on precisely one of the paths of the family. The path number, pn(G) is the minimal number of paths necessary to form a path decomposition of G .

On a generalization of transitivity for digraphs

Discrete Mathematics, 1988

In this paper we investigate the foliowing generalization of transitivity: A digraph D is (m, n)-transitive whenever there is a path of length m from x to y there is a subset of n + 1 vertices of these m + 1 vertices which contain a path of length n from x to y.