For a graph G and integer r ≥ 1 we denote the collection of independent r-sets of G by I (r) (G). If v ∈ V (G) then I (r) v (G) is the collection of all independent r-sets containing v. A graph G, is said to be r-EKR, for r ≥ 1, iff no intersecting family A ⊆ I (r) (G) is larger than max v∈V (G) |I (r) v (G)|. There are various graphs which are known to have his property: the empty graph of order n ≥ 2r (this is the celebrated Erdős-Ko-Rado theorem), any disjoint union of at least r copies of K t for t ≥ 2, and any cycle. In this paper we show how these results can be extended to other classes of graphs via a compression proof technique.In particular we show that any disjoint union of at least r complete graphs, each of order at least two, is r-EKR. We also show that paths are r-EKR for all r ≥ 1.

For a graph G vertex v of G and integer r 1, we denote the family of independent r-sets of V (G) by I (r) (G) and the subfamily {A ∈ I (r) (G): v ∈ A} by I (r) v (G); such a subfamily is called a star.Then, G is said to be r-EKR if no intersecting subfamily of I (r) (G) is larger than the largest star in I (r) (G). If every intersecting subfamily of I (r) v (G) of maximum size is a star, then G is said to be strictly r-EKR. We show that if a graph G is r-EKR then its lexicographic product with any complete graph is r-EKR.For any graph G, we define (G) to be the minimum size of a maximal independent vertex set. We conjecture that, if 1 r 1 2 (G), then G is r-EKR, and if r < 1 2 (G), then G is strictly r-EKR. This is known to be true when G is an empty graph, a cycle, a path or the disjoint union of complete graphs. We show that it is also true when G is the disjoint union of a pair of complete multipartite graphs.

Let S be a Steiner triple system and G a cubic graph. We say that G is S-colourable if its edges can be coloured so that at each vertex the incident colours form a triple of S: We show that if S is a projective system PGðn; 2Þ; nX2; then G is S-colourable if and only if it is bridgeless, and that every bridgeless cubic graph has an S-colouring for every Steiner triple system of order greater than 3. We establish a condition on a cubic graph with a bridge which ensures that it fails to have an S-colouring if S is an affine system, and we conjecture that this is the only obstruction to colouring any cubic graph with any non-projective system of order greater than 3. r

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