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October 19, 2025Open Access

Anti-Ramsey number of intersecting cliques

Key Points

The study establishes the anti-Ramsey number ar(n, F_{k, r}) for large n values, enhancing understanding of graph theory.
Specifically, the findings apply to values n ≥ 256r^{16}(k+1)^5, k ≥ 1, and r ≥ 2 in rainbow graphs.
The methodology focuses on determining the minimum number of colors needed to create rainbow copies of intersecting cliques.
Understanding these anti-Ramsey numbers may lead to advancements in combinatorial optimization and theoretical computer science.

Abstract

An edge-colored graph is called a rainbow graph if all its edges have distinct colors. The anti-Ramsey number ar (n, G), for a graph G and a positive integer n, is defined as the minimum number of colors r such that every exact r-edge-coloring of the complete graph Kₙ contains at least one rainbow copy of G. A (k, r) -fan graph, denoted F₊, ₑ, is a graph composed of k cliques each of size r, all intersecting at exactly one common vertex. In this paper, we determine ar (n, F₊, ₑ) for n 256r^16 (k+1) ⁵, k 1, and r 2.

Anti-Ramsey number of intersecting cliques

Key Points

Abstract

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