Imbeddings of Lexical Product of Graphs

It is our pleasure to welcome you at Bled, the site of the Third Slovenian Conference on Graph Theory. This conference has made a long way from its first meeting in Dubrovnik (now in Croatia) in 1985, organized by Tomaˇz Pisanski, the father of Graph Theory in Slovenia. The second meeting was held at Bled in 1991 and coincided with the outburst of the war for Slovenian independence. This caused a slight inconvenience to the 30 participants but the meeting will be remembered as a successful albeit adventurous event. This year the number of participants more than tripled. The received abstracts promise an interesting and fruitful contribution to mathematics. We express our thanks to all of you for attending this conference and wish you a mathematically productive week, but most of all a pleasant and relaxed stay in Slovenia. This collection contains only abstracts of the talks. The proceedings of the conference will be published as a special volume of Discrete Mathematics after thorough refereeing procedure following the standards of the journal. The organizers are grateful to all those who helped make this meeting possible

Notes on the independence number in the Cartesian product of graphs

Every connected graph G with radius r(G) and independence number α(G) obeys α(G) ≥ r(G). Recently the graphs for which equality holds have been classified. Here we investigate the members of this class that are Cartesian products. We show that for non-trivial graphs G and H, α(G ☐ H) = r(G ☐ H) if and only if one factor is a complete graph on two vertices, and the other is a nontrivial complete graph. We also prove a new (polynomial computable) lower bound α(G ☐ H) ≥ 2r(G)r(H) for the independence number and we classify graphs for which equality holds. The second part of the paper concerns independence irreducibility. It is known that every graph G decomposes into a König-Egervary subgraph (where the independence number and the matching number sum to the number of vertices) and an independence irreducible subgraph (where every non-empty independent set I has more than |I| neighbors). We examine how this decomposition relates to the Cartesian product. In particular, we show that if one of G or H is independence irreducible, then G ☐ H is independence irreducible