A note on two circumference generalizations of Chvátal's Hamiltonicity condition
A simple proof of Whitney's Theorem on connectivity in graphs
In 1932 Whitney showed that a graph with order is 2-connected if and only if any two vertices of are connected by at least two internally-disjoint paths. The above result and its proof have been used in some Graph Theory books, such as in Bondy and Murty’s well-known Graph Theory with Applications. In this note we give a much simple proof of Whitney’s Theorem.
A Sufficient Condition for a Graph to Be Hamiltonian.
A sufficient condition for Hamiltonian graphs
A traceability conjecture for oriented graphs.
A Triple of Heavy Subgraphs Ensuring Pancyclicity of 2-Connected Graphs
A graph G on n vertices is said to be pancyclic if it contains cycles of all lengths k for k ∈ {3, . . . , n}. A vertex v ∈ V (G) is called super-heavy if the number of its neighbours in G is at least (n+1)/2. For a given graph H we say that G is H-f1-heavy if for every induced subgraph K of G isomorphic to H and every two vertices u, v ∈ V (K), dK(u, v) = 2 implies that at least one of them is super-heavy. For a family of graphs H we say that G is H-f1-heavy, if G is H-f1-heavy for every graph...
A σ₃ type condition for heavy cycles in weighted graphs
A weighted graph is a graph in which each edge e is assigned a non-negative number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges. The weighted degree of a vertex v is the sum of the weights of the edges incident with v. In this paper, we prove the following result: Suppose G is a 2-connected weighted graph which satisfies the following conditions: 1. The weighted degree sum of any three independent vertices is at least m; 2. w(xz) = w(yz) for every...
Addendum to “Ring elements as sums of units”
We give a comment to Theorem 1.1 published in our paper “Ring elements as sums of units” [Cent. Eur. J. Math., 2009, 7(3), 395–399].
Alspach's problem: The case of Hamilton cycles and 5-cycles.
An alternative construction of normal numbers
A new class of -adic normal numbers is built recursively by using Eulerian paths in a sequence of de Bruijn digraphs. In this recursion, a path is constructed as an extension of the previous one, in such way that the -adic block determined by the path contains the maximal number of different -adic subblocks of consecutive lengths in the most compact arrangement. Any source of redundancy is avoided at every step. Our recursive construction is an alternative to the several well-known concatenative...
An Implicit Weighted Degree Condition For Heavy Cycles
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 G contains...
An upper bound for the shortest hamiltonian path in the symmetric euclidean case
An upper bound of a generalized upper Hamiltonian number of a graph
In this article we study graphs with ordering of vertices, we define a generalization called a pseudoordering, and for a graph we define the -Hamiltonian number of a graph . We will show that this concept is a generalization of both the Hamiltonian number and the traceable number. We will prove equivalent characteristics of an isomorphism of graphs and using -Hamiltonian number of . Furthermore, we will show that for a fixed number of vertices, each path has a maximal upper -Hamiltonian...
Arbitrarily traceable Eulerian graph has the Hamiltonian square
Balanced Gray codes.
Balancing cyclic -ary Gray codes.
Balancing cyclic -ary Gray codes. II.
Bent Hamilton cycles in -dimensional grid graphs.
Binary Gray codes with long bit runs.
Bipartite diametrical graphs of diameter 4 and extreme orders.