Lael F. Kinch

The bondage number of a graph

Abstract A set D of vertices in a graph G is a dominating set if each vertex of G that is not in D is adjacent to at least one vertex of D. The minimum cardinality among all dominating sets in G is called the domination number of G and denoted σ(G). We define the bondage number b(G) of a graph G to be the cardinality of a smallest set E of edges for which σ(G−E)>σ(G). Sharp bounds are obtained for b(G), and the exact values are determined for several classes of graphs.

On graphs having domination number half their order

In this paper we present a characterization of connected graphs of order 2n with domination numbern. Using this class of graphs, we determine an infinite class of graphs with the property that the domination number of the product of any two is precisely the product of the domination numbers.

On the domination of the products of graphs II: Trees

For a graph G, a subset of vertices D is a dominating set if for each vertex X not in D, X is adjacent to at least one vertex of D. The domination number, γ(G), is the order of the smallest such set. An outstanding conjecture in the theory of domination is for any two graph G and H, One result presented in this paper settles this question in the case when at least one of G or H is a tree. We show that for all graphs G and any tree T. Furthermore, we supply a partial characterization for which pairs of trees, T1 and T2, strict inequality occurs. We show for almost all pairs of trees.

The irregularity strength of tP 3

Abstract Let ( a 1 ,…, a t , b 1 ,…, b t ) be the sequence of distinct positive integers such that a i + b i are distinct for i = 1,…, t , and different from a j and b j , 1 ⩽ j ⩽ t . Denote by s ( t ) the minimum of the largest element of these sequences for fixed t . In this note we prove s ( t ) ⩾ ⌈(15 t − 1)/7⌉ and exhibit infinitely many sequences attaining equality. We also show s ( t ) ⩽ ⌈(15 t − 1)/7⌉ + 1 for every t . As a corollary we obtain that the irregularity strength of the graph G = tP 3 , the disjoint union of t paths of length 3, is about 5 n /7, where n = 3 t is the order of G.

Irregularity strength of dense graphs

Abstract It is proved that if t is a fixed positive integer and n is sufficiently large, then each graph of order n with minimum degree n − t has an assignment of weights 1, 2 or 3 to the edges in such a way that weighted degrees of the vertices become distinct.

On a generalization of transitivity for diagraphs

Abstract In this paper we investigate the following generalization of transitivity: A digraph D is ( m,n )- transitive whenever there is a path of length m from x to y there is a subset of n +1 vertices of these m +1 vertices which contain a path of length n from x to y . Here we study various properties of ( m,n )-transitive digraphs. In particular, ( m ,1)- transitive tournaments are characterized. Their similarities to transitive tournaments are analyzed and discussed. Various other results pertaining to ( m ,1)- transitive digraphs are given.

Sum-distinct sequences and Fibonacci numbers

Positive integers a, ~2” holds for every sumdistinct sequence. When one wants to create sum-distinct sequences A,, n = 1, 2, . . . , with small elements, the first thought consists in trying to extend A,-, with the smallest possible integer which is not the subsum of elements in AnO1. Not too surprisingly, this “first-fit” approach works when one starts with A, = {l},anditresultsinthesequenceof 2powersA, = (2’: 0 <i In 1) for each n. By using more sophisticated greedy procedures, one gets “better” sumdistinct sequences, that is sequences with smaller maximal element. It is worth noting that the sequence given by Conway and Guy in [2] (c.f. p. 64 in [6]), and which they conjecture answers the problem of ErdSs and Moser, is a greedy-like construction as well. The introduction of the irregularity strength of graphs by Chartrand et al. in [l] led to extend the concept of a sum-distinct sequence to more than one sequence and their generation by “greedy” procedures analogous to the first-fit approach.

RandomlyH-coverable graphs

AbstractA graph israndomly matchable if every matching of the graph is contained in a perfect matching. We generalize this notion and say that a graphG israndomly H-coverable if every set of independent subgraphs, each isomorphic toH, that does not cover the vertices ofG can be extended to a larger set of independent copies ofH. Various problems are considered for the situation whereH is a path. In particular, we characterize the graphs that are randomlyP3-coverable.