66 research outputs found
A note on Barnette's conjecture
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
On double domination in graphs
In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number γ ×2(G). A function f(p) is defined, and it is shown that γ ×2(G) = minf(p), where the minimum is taken over the n-dimensional cube Cn = {p = (p1,…,pn) | pi ∈ IR, 0 ≤ pi ≤ 1,i = 1,…,n}. Using this result, it is then shown that if G has order n with minimum degree δ and average degree d, then γ×2(G) ≤ ((ln(1+d)+lnδ+1)/δ)n
Random procedures for dominating sets in bipartite graphs
Using multilinear functions and random procedures, new upper bounds on the domination number of a bipartite graph in terms of the cardinalities and the minimum degrees of the two colour classes are established
New bounds on domination and independence in graphs
We propose new bounds on the domination number and on the independence number
of a graph and show that our bounds compare favorably to recent ones. Our
bounds are obtained by using the Bhatia-Davis inequality linking the variance,
the expected value, the minimum, and the maximum of a random variable with
bounded distribution
On domination in graphs
For a finite undirected graph G on n vertices two continuous optimization problems taken over the n-dimensional cube are presented and it is proved that their optimum values equal the domination number γ of G. An efficient approximation method is developed and known upper bounds on γ are slightly improved
A generalization of Tutte’s theorem on Hamiltonian cycles in planar graphs
AbstractIn 1956, W.T. Tutte proved that a 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. We prove that a planar graph G has a cycle containing a given subset X of its vertex set and any two prescribed edges of the subgraph of G induced by X if |X|≥3 and if X is 4-connected in G. If X=V(G) then Sanders’ result follows
On Selkow's bound on the independence number of graphs
For a graph G with vertex set V (G) and independence number α(G), Selkow [A Probabilistic lower bound on the independence number of graphs, Discrete Math. 132 (1994) 363–365] established the famous lower bound ∑v∈V(G)1d(v)+1(1+max{d(v)d(v)+1-∑u∈N(v)1d(u)+1,0}) on α (G), where N(v) and d(v) = |N(v)| denote the neighborhood and the degree of a vertex v ∈ V (G), respectively. However, Selkow’s original proof of this result is incorrect. We give a new probabilistic proof of Selkow’s bound here
Lightweight paths in graphs
Let k be a positive integer, G be a graph on V(G) containing a path on k vertices, and w be a weight function assigning each vertex v ∈ V(G) a real weight w(y). Upper bounds on the weight [formula] of P are presented, where P is chosen among all paths of G on k vertices with smallest weight
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