Let G be a graph with no three independent vertices. How many edges of G can be packed with edge-disjoint copies of Kk? More specifically, let fk(n,m) be the largest integer t such that for any graph with n vertices, m edges, and independence number 2, at least t edges can be packed with edge-disjoint copies ofKk. Turán’s Theorem together with Wilson’s Theorem assert that fk(n,m) = (1−o(1)) 2 4...

In this paper, we use coherent configurations to obtain new upper bounds on the independence number of orthogonal polarity graphs of projective planes of even order. In the case of classical planes of square even order, these bounds differ only by 1 from the size of the largest known independent sets.

We describe an approximation algorithm for the independence number of a graph. If a graph on n vertices has an independence number n/k + m for some fixed integer k ≥ 3 and some m > 0, the algorithm finds, in random polynomial time, an independent set of size Ω̃(m), improving the best known previous algorithm of Boppana and Halldorsson that finds an independent set of size Ω(m1/(k−1)) in such a g...

A random interval graph of order n is generated by picking 2n numbers X 1 : : : X 2n independently from the uniform distribution on 0; 1] and considering the collection of n intervals with extremities X 2i?1 and X 2i for i 2 f1;:::ng. The graph vertices correspond to intervals. Two vertices are connected if the corresponding intervals intersect. This paper characterizes the uctuations of the in...

The Chvátal–Erdős Theorem states that every graph whose connectivity is at least its independence number has a spanning cycle. In 1976, Fouquet and Jolivet conjectured an extension: If G is an n-vertex k-connected graph with independence number a, and a ≥ k, then G has a cycle with length at least k(n+a−k) a . We prove this conjecture.

The independence number of a graph G, denoted by α(G), is the maximum cardinality of an independent set of vertices in G. The transversal number of G is the minimum cardinality of a set of vertices that covers all the edges of G. If G is a bipartite graph of order n, then it is easy to see that n 2 ≤ α(G) ≤ n − 1. If G has no edges, then α(G) = n = n(G). Volkmann [Australas. J. Combin. 41 (2008...

Let G = (V,E) be a graph and k > 0 an integer. A k-independent set S ⊆ V is a set of vertices such that the maximum degree in the graph induced by S is at most k. With αk(G) we denote the maximum cardinality of a k-independent set of G. We prove that, for a graph G on n vertices and average degree d, αk(G) > k+1 ⌈d⌉+k+1n, improving the hitherto best general lower bound due to Caro and Tuza [Imp...

Since planar triangle-free graphs are 3-colourable, such a graph with n vertices has an independent set of size at least n/3. We prove that unless the graph contains a certain obstruction, its independence number is at least n/(3−ε) for some fixed ε > 0. We also provide a reduction rule for this obstruction, which enables us to transform any plane triangle-free graph G into a plane triangle-fre...

Aweakening of Hadwiger’s conjecture states that every n-vertex graphwith independence number α has a clique minor of size at least n α . Extending ideas of Fox (2010) [6], we prove that such a graph has a clique minor with at least n (2−c)α vertices where c > 1/19.2. © 2011 Elsevier B.V. All rights reserved.

For a connected and non-complete graph, a new lower bound on its independence number is proved. It is shown that this bound is realizable by the well known efficient algorithm MIN.