Upper Bounds on the Chromatic Index of Linear Hypergraphs

TL;DR

This paper addresses upper bounds on the chromatic index $q(V,E)$ of linear (loopless) hypergraphs and their connection to the Berge-Füredi conjecture, which predicts $q(V,E) \le \Delta([(V,E)]_{2}) + 1$. It develops two parallel strategies: a group-theoretic bound derived from color-preserving automorphisms via Burnside's lemma, and a combinatorial framework based on induced vertex colorings, equivalence relations, and Helly-type properties to connect $q(V,E)$ with the two-section graph degree and related quantities. The main contributions include a conceptual group-theoretic bound, a set of concrete combinatorial inequalities involving $\bar{C}^*$, $\sim$, $\theta$, and $\Gamma_{c_0}$, and two sufficient conditions (Helly-property of $H_{\Gamma}$ and a clique-based criterion for $k$-uniform linear hypergraphs) guaranteeing $q(V,E) \le \Delta([(V,E)]_{2})+1$. Together, these results provide a structural, framework-based route toward the Berge-Füredi conjecture and suggest avenues for further exploration of Boolean-function encoding, hypergraph-derived structures, and symmetry-based coloring bounds.

Abstract

We address the problem of finding upper bounds on the chromatic index $q(V,E)$ of linear (and loopless) hypergraphs. The first bound we find is defined through a color-preserving group on a proper and minimally edge-colored linear hypergraph, whose orbits serve as a finer partition to the hypergraph's coloring, thereby yielding an upper bound on $q(V,E)$. The next set of theorems in this paper relates to combinatorial properties of hypergraph coloring. Our results suggest a plausible approach to solving the Berge-Füredi conjecture, providing an upper bound on the chromatic index that directly relates $q(V,E)$ and $Δ([(V,E)]_{2}) + 1$. Furthermore, we provide three sufficient conditions for the conjecture to hold within this framework, when involving the Helly property for hypergraphs.

Figures (10)

• Figure 1: Group-theoretic bound example
• Figure 2: A linear hypergraph $(V,E), \bar{C}$ with its proper minimal coloring of 7 colors.
• Figure 3: This graph represents $V/\sim$ with its equivalence classes as vertices.
• Figure 4: This subgraph shows $\Omega/\theta$, with the two vertices defining $\Gamma_{7}$ highlighted in dark blue.
• Figure 5: This is the hypergraph (just a graph in this case) $H_{\Gamma_{7}}$, whose vertices are colors

Theorems & Definitions (46)

• Conjecture 1.1: Berge-Füredi conjecture
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Proposition 2.7
• proof