The independence number of graphs with large odd girth

Discrete Mathematics, 2008

Murty [A generalization of the Hoffman-Singleton graph, Ars Combin. 7 (1979) 191-193.] constructed a family of (p m + 2)regular graphs of girth five and order 2p 2m , where p 5 is a prime, which includes the Hoffman-Singleton graph [A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. (1960) 497-504]. This construction gives an upper bound for the least number f (k) of vertices of a k-regular graph with girth 5. In this paper, we extend the Murty construction to k-regular graphs with girth 5, for each k. In particular, we obtain new upper bounds for f (k), k 16.

2012

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domatic number of a graph G is the maximum number of total dominating sets into which the vertex set of G can be partitioned. We show that the total domatic number of a random r-regular graph is almost surely at most r − 1, and that for 3-regular random graphs, the total domatic number is almost surely equal to 2. We also give a lower bound on the total domatic number of a graph in terms of order, minimum degree and maximum degree. As a corollary, we obtain the result that the total domatic number of an r-regular graph is at least r/(3 ln(r)). MSC(2010): 05C69.

2011

The independent domination number i(G) (independent number β(G)) is the minimum (maximum) cardinality among all maximal independent sets of G. conjectured that any connected regular graph G of order n and degree δ 2n/3δ 1 2 δ. For 1 k l m, the subset graph Sm(k, l) is the bipartite graph whose vertices are the kand l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. In this paper, we give a sharp upper bound for i(Sm(k, l)) and prove that if k + l = m then Haviland's conjecture holds for the subset graph Sm(k, l). Furthermore, we give the exact value of β(Sm(k, l)).

Discrete Mathematics, 2013

We prove that for integers r and D with r ≥ 2 and D ≥ 3, there are only finitely many connected graphs of minimum degree at least 2, maximum degree at most D, and girth at least 7 that have maximal independent sets of at most r different sizes. Furthermore, we prove several results restricting the degrees of such graphs. Our contributions generalize known results on well-covered graphs.

Israel Journal of Mathematics, 1964

Lower and upper bounds for the maximal number of independent vertices in a regular graph are obtained, it is shown that the bounds are best possible. Some properties of regular graphs concerning the property .~ defined below are investigated.

Ars Combinatoria - ARSCOM, 2004

The domatic number of a graph G is the maximum number of dominating sets into which the vertex set of G can be partitioned. We show that the domatic number of a random r-regular graph is almost surely at most r, and that for 3-regular random graphs, the domatic number is almost surely equal to 3. We also give a lower bound on the domatic number of a graph in terms of order, minimum degree and maximum degree. As a corol- lary, we obtain the result that the domatic number of an r-regular graph is at least (r + 1)/(3ln(r + 1)).

Graphs and Combinatorics, 1992

Let r be a positive integer. Consider r-regular graphs in which no induced subgraph on four vertices is an independent pair of edges. The number v of vertices in such a graph does not exceed 5r/2; this proves a conjecture of Bermond. More generally, it is conjectured that if v > 2r, then the ratio v/r must be a rational number of the form 2 + 1/(2k). This is proved for v/r > 21 _~. The extremal graphs and many other classes of these graphs are described and characterized.

Discrete Mathematics, 2019

A (k, g)-graph is a k-regular graph with girth g and a (k, g)-cage is a (k, g)-graph with the fewest possible number of vertices n(k, g). Constructing (k, g)-cages and determining the order are both very hard problems. For this reason, an intensive line of research is devoted to constructing smaller (k, g)-graphs than previously known ones, providing in this way new upper bounds to n(k, g) each time such a graph is constructed. The paper focuses on girth g = 5, where cages are known only for degrees k ≤ 7. We construct (k, 5)-graphs using and extending techniques of amalgamation into the incidence graphs of elliptic semiplanes of type L introduced and exposed by . The order of these graphs provides better upper bounds on n(k, 5) than those known so far, for values of k such that either 13 ≤ k ≤ 33 or k ≥ 66.

Discrete Mathematics, 2019

This paper generalizes and unifies the existing spectral bounds on the k-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than k. The previous bounds known in the literature follow as a corollary of the main results in this work. We show that for most cases our bounds outperform the previous known bounds. Some infinite graphs where the bounds are tight are also presented. Finally, as a byproduct, we derive some lower spectral bounds for the diameter of a graph.

Arxiv preprint math/0310379, 2003

We enumerate the independent sets of several classes of regular and almost regular graphs and compute the corresponding generating functions. We also note the relations between these graphs and other combinatorial objects and, in some cases, construct the corresponding bijections.