68 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

### 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})$\sum {_{v \in V(G)}{1 \over {d(v) + 1}}} \left( {1 + \max \left\{ {{{d(v)} \over {d(v) + 1}} - \sum {_{u \in N(v)}{1 \over {d(u) + 1}},0} } \right\}} \right)$ 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

### 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

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