A Proof of the Erd\"os - Faber - Lov\'asz Conjecture

TL;DR

This paper provides an algorithmic proof of the Erd"os-Faber-Lovász conjecture by utilizing symmetric Latin squares and analyzing clique degrees in specific graph structures.

Contribution

It introduces a novel algorithmic approach to prove the conjecture, connecting Latin squares with graph coloring and clique degrees.

Findings

Proof of the conjecture using symmetric Latin squares

Development of an algorithmic coloring method for the graph class

Establishment of a link between clique degrees and coloring strategies

Abstract

In 1972, Erd\"{o}s - Faber - Lov\'{a}sz (EFL) conjectured that, if $\textbf{H}$ is a linear hypergraph consisting of $n$ edges of cardinality $n$, then it is possible to color the vertices with $n$ colors so that no two vertices with the same color are in the same edge. In 1978, Deza, Erd\"{o}s and Frankl had given an equivalent version of the same for graphs: Let $G= \bigcup _{i=1}^{n} A_i$ denote a graph with $n$ complete graphs $A_1, A_2,$ $ \dots , A_n$, each having exactly $n$ vertices and have the property that every pair of complete graphs has at most one common vertex, then the chromatic number of $G$ is $n$. The clique degree $d^K(v)$ of a vertex $v$ in $G$ is given by $d^K(v) = |\{A_i: v \in V(A_i), 1 \leq i \leq n\}|$. In this paper we give an algorithmic proof of the conjecture using the symmetric latin squares and clique degrees of the vertices of $G$.