Rainbow numbers for graphs with cyclomatic number at most two

TL;DR

This paper investigates rainbow numbers in edge-coloured complete graphs, classifying them based on the graph's cyclomatic number, and computes specific rainbow numbers for certain graphs like the bull, diamond, K_2,3, and house.

Contribution

It provides a classification of rainbow numbers according to cyclomatic number and explicitly computes these numbers for several specific graphs.

Findings

Rainbow numbers grow unboundedly for graphs with cyclomatic number ≥ 2.

Rainbow numbers are linear in n for graphs with cyclomatic number 1.

Explicit rainbow numbers are computed for the bull, diamond, K_2,3, and house graphs.

Abstract

For a given graph H and n ? 1; let f(n;H) denote the maximum number m for which it is possible to colour the edges of the complete graph Kn with m colours in such a way that each subgraph H in Kn has at least two edges of the same colour. Equivalently, any edge-colouring of Kn with at least rb(n;H) = f(n;H)+1 colours contains a rainbow copy of H: The numbers f(n;H) and rb(Kn;H) are called anti-ramsey numbers and rainbow numbers, respectively. In this paper we will classify the rainbow number for a given graph H with respect to its cyclomatic number. Let H be a graph of order p >= 4 and cyclomatic number v(H) >= 2: Then rb(Kn;H) cannot be bounded from above by a function which is linear in n: If H has cyclomatic number v(H) = 1; then rb(Kn;H) is linear in n: We will compute all rainbow numbers for the bull B; which is the unique graph with 5 vertices and degree sequence (1; 1; 2; 3; 3): Furthermore, we will compute some rainbow numbers for the diamond D = K_4 - e; for K_2;3; and for the house H (complement of P_5).