An improved bound for the strong clique index of graphs

Abstract: For a graph $G$ with line graph $L(G)$, $χ(L(G)<sup>2)$ and $ω(L(G)<sup>2)$ are called the \emph{strong chromatic index} and \emph{strong clique index} of $G$, respectively. A well-known conjecture of Erdős and Nešetřil (1985) posits that $χ(L(G)^2)\le \frac{5}{4}Δ(G)<sup>2$. Related to that, Faudree, Gyárfás, Schelp and Tuza (1990) conjectured that $ω(L(G)²⁾ \le \frac{5}{4}Δ(G)<sup>2$. We show that $ω(L(G)²⁾ \le \frac{2607}{1987}Δ(G)² &lt; \frac{21}{16}Δ(G)<sup>2$ improving the upper bound $\frac{4}{3}Δ(G)<sup>2$ of Faron and Postle. Indeed, we make progress towards a stronger conjecture of Faron and Postle in terms of Ore-degree. For positive integers $Δ$ and $t$, let $h_t(Δ)$ denote the smallest integer such that any graph $G$ with size at least $h_t(Δ)$ and maximum degree $Δ(G)\le Δ$, contains two edges with distance at least $t$. An old problem of Erdős and Nešetřil (1986) concerns estimating the quantity $h_t(Δ)$ and can be thought of as the edge-version of the degree-diameter problem. Chung, Gyárfás, Tuza and Trotter established the sharp inequality $h_2(Δ)\le \frac{5}{4}Δ<sup>2+1$. We disprove two conjectures of Cambie, Cames van Batenburg, Joannis de Verclos and Kang concerning the next open case $h_3(Δ)$.

• The paper shows that ω(L(G)²) is bounded by (2607/1987)Δ(G)², significantly advancing prior estimates toward the conjectured optimum.
• It introduces sharper bounds by leveraging the Ore-degree and a detailed inductive analysis on bipartite subgraphs to refine clique estimates.
• The authors construct infinite families of graphs that serve as counterexamples to recent conjectures on edge-degree-diameter properties.

Improved Upper Bounds for the Strong Clique Index of Graphs

Introduction

The strong clique index, denoted $\omega(L(G)^2)$ for a graph $G$, captures the largest size of an edge set inducing a clique in the square of the line graph of $G$. It is tightly connected to longstanding open problems related to the strong chromatic index $\chi(L(G)^2)$, following the seminal conjecture of Erdős and Nešetřil that both indices should be bounded above by $\frac{5}{4}\Delta(G)^2$ in terms of the maximum degree $\Delta(G)$. This paper provides new progress on improving upper bounds for $\omega(L(G)^2)$, introduces sharper estimates involving the Ore-degree, and constructs infinite families of counterexamples to recent conjectures on edge-degree-diameter numbers.

Overview of Main Results

The principal contribution is the establishment of the bound

$$\omega(L(G)^2) \leq \frac{2607}{1987} \Delta(G)^2 < \frac{21}{16}\Delta(G)^2,$$

which supersedes the prior state-of-the-art bound $\omega(L(G)^2) \leq \frac{4}{3}\Delta(G)^2$ due to Faron and Postle. This result advances the field toward the target bound of $\frac{5}{4}\Delta(G)^2$ and aligns with the broader conjectural landscape regarding clique numbers in squares of line graphs.

Moreover, the paper achieves non-trivial partial progress on a stronger Ore-degree based conjecture of Faron and Postle involving bipartite subgraphs $G$0 of $G$1 for which $G$2 forms a clique in $G$3. Specifically, it proves

$G$4

where $G$5 is the maximum Ore-degree of $G$6 in $G$7. This result is leveraged inductively to deliver the improved global bound on the strong clique index.

Technical Approach

The derivation of the new bounds is based on a sophisticated inductive analysis, which proceeds via several technical lemmas relating edge sets in bipartite subgraphs and their interaction with the distance structure in $G$8. The proof strategy involves:

• Careful partitioning of vertex neighborhoods and induced subgraphs to manage the counting of edges incident with chosen vertex parts.
• Detailed recursive analysis on the sizes of induced cliques in $G$9, informed by bounds on Ore-degrees and structural properties of bipartite subgraphs.
• Quadratic inequality analysis and explicit polynomial estimates to pin down critical parameters, ensuring that the improved constants can be certified without reliance on non-elementary computation.

The choice to work with Ore-degree rather than simply maximum degree is crucial—it captures finer combinatorial information about local edge distribution and enables sharpening of the inductive hypothesis.

Degree-Diameter-Type Problems and Counterexamples

Beyond global upper bounds, the paper addresses the quantity $G$0, a fundamental extremal parameter related to the edge-degree-diameter problem: it is the largest possible size of a graph of maximum degree at most $G$1 whose line graph's $G$2-th power is a complete graph. Tight bounds on $G$3 are known only for small $G$4, with the case $G$5 remaining open.

The authors construct explicit infinite families of graphs—arising from blowups of distance-regular graphs and manipulations of polarity graphs (notably via the projective plane $G$6)—that provide counterexamples to two recent conjectures concerning $G$7. In particular, the result

$G$8

disproves the conjectured upper bound $G$9 for all sufficiently large $\chi(L(G)^2)$0, revealing a previously unappreciated gap.

The construction leverages properties of the Odd graph $\chi(L(G)^2)$1 and the truncated Witt graph, whose line graph cube is complete due to high connectivity and tailor-made combinatorial structure.

Numerical Improvements and Comparative Discussion

The new bound $\chi(L(G)^2)$2 significantly lowers the multiplicative constant in the strong clique index relative to previous results ($\chi(L(G)^2)$3), moving the state-of-the-art closer to the conjectured optimal constant $\chi(L(G)^2)$4.

Furthermore, the infinite family of counterexamples shifts the understanding of degree-diameter-type growth rates for $\chi(L(G)^2)$5, imposing a new lower limit of $\chi(L(G)^2)$6 for the coefficient in the leading term of $\chi(L(G)^2)$7. This provides an obstruction to certain optimistic conjectures about asymptotic sharpness in this regime.

Implications and Future Directions

The theoretical implications of these results are twofold:

1. Combinatorial Tightening: The incremental improvement in the multiplicative constant for $\chi(L(G)^2)$8 suggests the need for fundamentally new tools or techniques to push further toward $\chi(L(G)^2)$9, as the current combinatorial technology appears to plateau near $\frac{5}{4}\Delta(G)^2$0.
2. Structural Barriers: The explicit construction of families of graphs defying previously held conjectures on $\frac{5}{4}\Delta(G)^2$1 indicates that the growth of edge-complete subgraphs in line graph powers can be more robust than previously predicted, especially when informed by deep structures from extremal and algebraic graph theory (e.g., projective planes and polarity graphs).

Practically, these results influence the extremal design of network topologies, coloring and partition strategies in communication networks, and the analysis of algorithmic coloring heuristics, especially where edge constraints and clique-like substructures dominate.

Theoretically, further exploration of Ore-degree-centric bounds, extensions to higher $\frac{5}{4}\Delta(G)^2$2 in edge-degree-diameter problems, and deeper analysis of the interplay between local and global edge distributions appear promising. Moreover, sharpening the constant below $\frac{5}{4}\Delta(G)^2$3 is posited to require fundamentally new ideas.

Conclusion

This work delivers a notable advance in bounding the strong clique index of graphs by producing an explicit improved upper bound, with rigorous inductive and combinatorial arguments centered on Ore-degree analysis. It also invalidates conjectures on edge-degree-diameter parameters for line graph powers by constructing explicit infinite counterexample families. These results have both immediate extremal combinatorics significance and point the way to several challenging open questions in the structure and extremal theory of graph powers.

Reference: "An improved bound for the strong clique index of graphs" (2607.02698)