Let graph G = (V,E) and integer b ≥ 1 be given. A set S ⊆ V is said to be b-independent if u, v ∈ S implies dG(u, v) > b where dG(u, v) is the shortest distance between u and v in G. The b-independence number αb(G) is the size of the largest b-independent subset of G. When b = 1 this reduces to the standard definition of independence number. We study this parameter in relation to the random gra...

We present a lower bound on the independence number of arbitrary hypergraphs in terms of the degree vectors. The degree vector of a vertex v is given by d(v) = (d 1 (v); d 2 (v); : : :) where d m (v) is the number of edges of size m containing v. We deene a function f with the property that any hypergraph H = (V; E) satisses (H) P v2V f(d(v)). This lower bound is sharp when H is a matching, and...

Ohba’s conjecture states that if a graph G has chromatic number χ(G) = k and has at most 2k+1 vertices, then G has choice number Ch(G) equal to χ(G). We prove that Ohba’s conjecture is true for each graph that has independence number 3. We also prove that Ohba’s conjecture is true for each graph G that has an independent set S of size 4 such that G\S has independence number 3.

Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs shown in Figure 1 as a subgraph. We prove that the independence number of G is at least n(G)/3 + nt(G)/42, where n(G) is the number of vertices in G and nt(G) is the number of nontriangle vertices in G. This bound is tight as implied by the wellknown tight lower bound of 5n(G)/14 on t...

We study the reachability problem for finite directed graphs whose independence number is bounded by some constant k. This problem is a generalisation of the reachability problem for tournaments. We show that the problem is first-order definable for all k. In contrast, the reachability problems for many other types of finite graphs, including dags and trees, are not first-order definable. Also ...

A 2-factor-plus-triangles graph is the union of two 2-regular graphs G1 and G2 with the same vertices, such that G2 consists of disjoint triangles. Let G be the family of such graphs. These include the famous “cycle-plus-triangles” graphs shown to be 3-choosable by Fleischner and Stiebitz. The independence ratio of a graph in G may be less than 1/3; but achieving the minimum value 1/4 requires ...

We explore the relationship between Kekulé structures and maximum face independence sets in fullerenes: plane trivalent graphs with pentagonal and hexagonal faces. For the class of leap-frog fullerenes, we show that a maximum face independence set corresponds to a Kekulé structure with a maximum number of benzene rings and may be constructed by partitioning the pentagonal faces into pairs and 3...

The independence number of a graph is defined as the maximum size of a set of pairwise non-adjacent vertices and the spectral radius is defined as the maximum eigenvalue of the adjacency matrix of the graph. Xu et al. in [The minimum spectral radius of graphs with a given independence number, Linear Algebra and its Applications 431 (2009) 937–945] determined the connected graphs of order n with...

An even order graph G with a perfect matching is k-extendable if for every matching M of size k in G, there exists a perfect matching in G containing all the edges of M. In this note, we estabUsh a necessary and sufficient condition for a graph to be k-extendable in terms of its independence number. All graphs considered in this note are finite, connected, loopless and have no multiple edges. F...

In this paper, we study spectral versions of the densest subgraph problem and the largest independence subset problem. In the first part, we give an algorithm for identifying small subgraphs with large spectral radius. We also prove a Hoffman-type ratio bound for the order of an induced subgraph whose spectral radius is bounded from above.