Displaying similar documents to “On constant-weight TSP-tours”

Heavy cycles in weighted graphs

J. Adrian Bondy, Hajo J. Broersma, Jan van den Heuvel, Henk Jan Veldman (2002)

Discussiones Mathematicae Graph Theory

Similarity:

An (edge-)weighted graph is a graph in which each edge e is assigned a nonnegative real number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges, and an optimal cycle is one of maximum weight. The weighted degree w(v) of a vertex v is the sum of the weights of the edges incident with v. The following weighted analogue (and generalization) of a well-known result by Dirac for unweighted graphs is due to Bondy and Fan. Let G be a 2-connected weighted...

Edge cycle extendable graphs

Terry A. McKee (2012)

Discussiones Mathematicae Graph Theory

Similarity:

A graph is edge cycle extendable if every cycle C that is formed from edges and one chord of a larger cycle C⁺ is also formed from edges and one chord of a cycle C' of length one greater than C with V(C') ⊆ V(C⁺). Edge cycle extendable graphs are characterized by every block being either chordal (every nontriangular cycle has a chord) or chordless (no nontriangular cycle has a chord); equivalently, every chord of a cycle of length five or more has a noncrossing chord.

Pancyclicity when each Cycle Must Pass Exactly k Hamilton Cycle Chords

Fatima Affif Chaouche, Carrie G. Rutherford, Robin W. Whitty (2015)

Discussiones Mathematicae Graph Theory

Similarity:

It is known that Θ(log n) chords must be added to an n-cycle to produce a pancyclic graph; for vertex pancyclicity, where every vertex belongs to a cycle of every length, Θ(n) chords are required. A possibly ‘intermediate’ variation is the following: given k, 1 ≤ k ≤ n, how many chords must be added to ensure that there exist cycles of every possible length each of which passes exactly k chords? For fixed k, we establish a lower bound of ∩(n1/k) on the growth rate.

An Implicit Weighted Degree Condition For Heavy Cycles

Junqing Cai, Hao Li, Wantao Ning (2014)

Discussiones Mathematicae Graph Theory

Similarity:

For a vertex v in a weighted graph G, idw(v) denotes the implicit weighted degree of v. In this paper, we obtain the following result: Let G be a 2-connected weighted graph which satisfies the following conditions: (a) The implicit weighted degree sum of any three independent vertices is at least t; (b) w(xz) = w(yz) for every vertex z ∈ N(x) ∩ N(y) with xy /∈ E(G); (c) In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight. Then...