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Izolda Gorgol

Abstract

We say that a graph F strongly arrows a pair of graphs (G,H) and write F ind(G,H) if any 2-coloring of its edges with red and blue leads to either a red G or a blue H appearing as induced subgraphs of F. The induced Ramsey number, IR(G,H) is defined as min{|V (F)| : F ind (G,H)}. We will consider two aspects of induced Ramsey numbers. Firstly we will show that the lower bound of the induced Ramsey number for a connected graph G with independence number α and a graph H with clique number ω is roughly ω2α2. This bound is sharp. Moreover we will also consider the case when G is not connected providing also a sharp lower bound which is linear in both parameters.

Open access

Izolda Gorgol and Anna Lechowska

Abstract

Let ar(G,H) be the largest number of colors such that there exists an edge coloring of G with ar(G,H) colors such that each subgraph isomorphic to H has at least two edges in the same color. We call ar(G,H) the anti- Ramsey number for a pair of graphs (G,H). This notion was introduced by Erdős, Simonovits and Sόs in 1973 and studied in numerous papers. Hanoi graphs were introduced by Scorer, Grundy and Smith in 1944 as the model of the well known Tower of Hanoi puzzle. In the paper we study the anti-Ramsey number of Hanoi graphs and consider them both as the graph G and H. Among others we present the exact value of the anti-Ramsey number in case when both graphs are constructed for the same number of pegs.