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Displaying 21 – 40 of 71

A Note on a Bermond's Conjecture

Danut Marcu (1986)

Publications de l'Institut Mathématique

A note on a new condition implying pancyclism

Evelyne Flandrin, Hao Li, Antoni Marczyk, Mariusz Woźniak (2001)

Discussiones Mathematicae Graph Theory

We first show that if a 2-connected graph G of order n is such that for each two vertices u and v such that δ = d(u) and d(v) < n/2 the edge uv belongs to E(G), then G is hamiltonian. Next, by using this result, we prove that a graph G satysfying the above condition is either pancyclic or isomorphic to $K_{n / 2, n / 2}$ .

A note on arc-disjoint cycles in tournaments

Jan Florek (2014)

Colloquium Mathematicae

We prove that every vertex v of a tournament T belongs to at least $m a x m i n δ ⁺ (T), 2 δ ⁺ (T) - d ⁺_{T} (v) + 1, m i n δ ¯ (T), 2 δ ¯ (T) - d ¯_{T} (v) + 1$ arc-disjoint cycles, where δ⁺(T) (or δ¯(T)) is the minimum out-degree (resp. minimum in-degree) of T, and $d ⁺_{T} (v)$ (or $d ¯_{T} (v)$ ) is the out-degree (resp. in-degree) of v.

A Note on Barnette’s Conjecture

Jochen Harant (2013)

Discussiones Mathematicae Graph Theory

Barnette conjectured that each planar, bipartite, cubic, and 3-connected graph is hamiltonian. We prove that this conjecture is equivalent to the statement that there is a constant c > 0 such that each graph G of this class contains a path on at least c|V (G)| vertices.

A note on circuit graphs.

Cui, Qing (2010)

The Electronic Journal of Combinatorics [electronic only]

A note on cycle double covers in Cayley graphs.

Hoffman, F., Locke, S.C., Meyerowitz, A.D. (1991)

Mathematica Pannonica

A note on dimension of $P_{3}^{n}$

Svatopluk Poljak, Daniel Turzík (1981)

Czechoslovak Mathematical Journal

A note on graphs without short even cycles.

Lam, Thomas, Verstraëte, Jacques (2005)

The Electronic Journal of Combinatorics [electronic only]

A Note on Longest Paths in Circular Arc Graphs

Felix Joos (2015)

Discussiones Mathematicae Graph Theory

As observed by Rautenbach and Sereni [SIAM J. Discrete Math. 28 (2014) 335-341] there is a gap in the proof of the theorem of Balister et al. [Combin. Probab. Comput. 13 (2004) 311-317], which states that the intersection of all longest paths in a connected circular arc graph is nonempty. In this paper we close this gap.

A note on minimally 3-connected graphs

Víctor Neumann-Lara, Eduardo Rivera-Campo, Jorge Urrutia (2004)

Discussiones Mathematicae Graph Theory

If G is a minimally 3-connected graph and C is a double cover of the set of edges of G by irreducible walks, then |E(G)| ≥ 2| C| - 2.

A note on odd cycle-complete graph Ramsey numbers.

Sudakov, Benny (2002)

The Electronic Journal of Combinatorics [electronic only]

A Note on Path Domination

Liliana Alcón (2016)

Discussiones Mathematicae Graph Theory

We study domination between different types of walks connecting two non-adjacent vertices u and v of a graph (shortest paths, induced paths, paths, tolled walks). We succeeded in characterizing those graphs in which every uv-walk of one particular kind dominates every uv-walk of other specific kind. We thereby obtained new characterizations of standard graph classes like chordal, interval and superfragile graphs.

A note on removing a point of a strong digraph

Peter Horák (1983)

Mathematica Slovaca

A note on the circumference of graphs.

Stacho, L. (1995)

Acta Mathematica Universitatis Comenianae. New Series

A note on the number of Hamiltonian paths in strong tournaments.

Busch, Arthur H. (2006)

The Electronic Journal of Combinatorics [electronic only]

A note on the Song-Zhang theorem for Hamiltonian graphs

Kewen Zhao, Ronald J. Gould (2010)

Colloquium Mathematicae

An independent set S of a graph G is said to be essential if S has a pair of vertices that are distance two apart in G. In 1994, Song and Zhang proved that if for each independent set S of cardinality k+1, one of the following condition holds: (i) there exist u ≠ v ∈ S such that d(u) + d(v) ≥ n or |N(u) ∩ N(v)| ≥ α (G); (ii) for any distinct u and v in S, |N(u) ∪ N(v)| ≥ n - max{d(x): x ∈ S}, then G is Hamiltonian. We prove that if for each essential...

A Note on the Total Detection Numbers of Cycles

Henry E. Escuadro, Futaba Fujie, Chad E. Musick (2015)

Discussiones Mathematicae Graph Theory

Let G be a connected graph of size at least 2 and c :E(G)→{0, 1, . . . , k− 1} an edge coloring (or labeling) of G using k labels, where adjacent edges may be assigned the same label. For each vertex v of G, the color code of v with respect to c is the k-vector code(v) = (a0, a1, . . . , ak−1), where ai is the number of edges incident with v that are labeled i for 0 ≤ i ≤ k − 1. The labeling c is called a detectable labeling if distinct vertices in G have distinct color codes. The value val(c) of...

A note on two circumference generalizations of Chvátal's Hamiltonicity condition

Ladislav Stacho (2002)

Mathematica Slovaca

A path(ological) partition problem

Izak Broere, Michael Dorfling, Jean E. Dunbar, Marietjie Frick (1998)

Discussiones Mathematicae Graph Theory

Let τ(G) denote the number of vertices in a longest path of the graph G and let k₁ and k₂ be positive integers such that τ(G) = k₁ + k₂. The question at hand is whether the vertex set V(G) can be partitioned into two subsets V₁ and V₂ such that τ(G[V₁] ) ≤ k₁ and τ(G[V₂] ) ≤ k₂. We show that several classes of graphs have this partition property.

A proof of some Schützenberger-type results for Eulerian paths and circuits on digraphs.

Chwe, Byoung-Song (1994)

International Journal of Mathematics and Mathematical Sciences

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