# SOME REMARKS ON THE STRUCTURE OF STRONG k-TRANSITIVE DIGRAPHS

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- New Ore's Type Results on Hamiltonicity and Existence of Paths of Given Length in Graphs. Lichiardopol, Nicolas // Graphs & Combinatorics;Jan2013, Vol. 29 Issue 1, p99
The well-known Ore's theorem (see Ore in Am Math Mon 65:55, ), states that a graph G of order n such that d( x) + d( y) â‰¥ n for every pair { x, y} of non-adjacent vertices of G is Hamiltonian. In this paper, we considerably improve this theorem by proving that in a graph G of order n and...

- An Algorithm to Detect Cycle in an Undirected Graph. Kumar, Anand; Jani, N. N. // International Journal of Computational Intelligence Research;2010, Vol. 6 Issue 2, p305
This paper presents a novel algorithm to detect cycles in a graph. The graph may be of any type. Cycles are available in a graph and in much real life application; it is required to know the existence of cycles in a graph. This algorithm is developed in the context of network design problem but...

- Neighborhood Unions for the Existence of Disjoint Chorded Cycles in Graphs. Gao, Yunshu; Li, Guojun; Yan, Jin // Graphs & Combinatorics;Sep2013, Vol. 29 Issue 5, p1337
A chorded cycle is a cycle with at least one chord. We prove that if G is a simple graph with order n â‰¥ 4 k and $${|N_G(x)\cup N_G(y)|\geq 4k+1}$$ for each nonadjacent pair of vertices x and y, then G contains k vertex-disjoint chorded cycles. The degree condition is sharp in general.

- UNDERLYING GRAPHS OF 3-QUASI-TRANSITIVE DIGRAPHS AND 3-TRANSITIVE DIGRAPHS RUIXIA WANG, SHIYING WANG. RUIXIA WANG; SHIYING WANG // Discussiones Mathematicae: Graph Theory;2013, Vol. 33 Issue 2, p429
A digraph is 3-quasi-transitive (resp. 3-transitive), if for any path x0x1 x2x3 of length 3, x0 and x3 are adjacent (resp. x0 dominates x3). CÃ©sar HernÃ¡ndez-Cruz conjectured that if D is a 3-quasi-transitive digraph, then the underlying graph of D, UG(D), admits a 3-transitive orientation....

- On Sullivan's conjecture on cycles in 4-free and 5-free digraphs. Liang, Hao; Xu, Jun // Acta Mathematica Sinica;Jan2013, Vol. 29 Issue 1, p53
For a simple digraph G, let Î²( G) be the size of the smallest subset X âŠ† E( G) such that Gâˆ’X has no directed cycles, and let Î³( G) be the number of unordered pairs of nonadjacent vertices in G. A digraph G is called k-free if G has no directed cycles of length at most k. This...

- A Contribution to the Second Neighborhood Problem. Ghazal, Salman // Graphs & Combinatorics;Sep2013, Vol. 29 Issue 5, p1365
Seymour's Second Neighborhood Conjecture asserts that every oriented graph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. It is proved for tournaments, tournaments missing a matching and tournaments missing a generalized star. We...

- Improved Sufficient Conditions for the Existence of Anti-Directed Hamiltonian Cycles in Digraphs. Busch, Arthur; Jacobson, Michael; Morris, Timothy; Plantholt, Michael; Tipnis, Shailesh // Graphs & Combinatorics;May2013, Vol. 29 Issue 3, p359
Let D be a directed graph of order n. An anti-directed ( hamiltonian) cycle H in D is a (hamiltonian) cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. In this paper we give sufficient conditions for the existence of anti-directed hamiltonian...

- Î³-CYCLES AND TRANSITIVITY BY MONOCHROMATIC PATHS IN ARC-COLOURED DIGRAPHS. CASAS-BAUTISTA, ENRIQUE; GALEANA-SÁNCHEZ, HORTENSIA; ROJAS-MONROY, ROCÍO // Discussiones Mathematicae: Graph Theory;2013, Vol. 33 Issue 3, p493
We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. If D is an m-coloured digraph and a âˆˆ A(D), colour(a) will denote the colour has been used on a. A path (or a cycle) is called monochromatic if all of its arcs are coloured alike. A Î³-cycle in D is a...

- The Existence of an Alternating Sign on a Spanning Tree of Graphs. KIM, DONGSEOK; KWON, YOUNG SOO; LEE, JAEUN // Kyungpook Mathematical Journal;Dec2012, Vol. 52 Issue 4, p513
For a spanning tree T of a connected graph .. and for a labelling Î¦: E(T ) ! {+,-}, Î¦ is called an alternating sign on a spanning tree T of a graph ... if for any cotree edge e âˆˆ E(...)-E(T ), the unique path in T joining both end vertices of e has alternating signs. In the present...