Proper edge coloring with rainbow diamonds

Abstract: Motivated by the B-coloring defined by Gyárfás and Sárközy, we introduce a new edge coloring called \emph{D-coloring}. For a graph $G$, a D-coloring of $G$ is a proper edge coloring such that every diamond subgraph is rainbow. The \emph{D-chromatic index} of $G$, denoted by $χ'_D(G)$, is the minimum number of colors needed for a D-coloring of $G$. Denote by $Δ$ the maximum degree of $G$. We prove that $χ'_D(G)\le \frac{9}{16}Δ^2+\frac{1}{2}Δ$, conjecture that $χ'_D(G)\le \frac{1}{2}Δ^2+\frac{1}{2}Δ$, and verify this conjecture for $Δ\le 5$.

• The paper establishes a novel D-coloring paradigm that guarantees every diamond subgraph is rainbow, positioning it between proper and strong edge colorings.
• The paper proves a general upper bound of (9/16)Δ² + (1/2)Δ for the D-chromatic index and confirms the conjectured optimal bound for graphs with Δ ≤ 5.
• The paper explores implications for planar graphs and resource allocation, while motivating further research on algorithmic approaches and structural extensions.

Proper Edge Coloring with Rainbow Diamonds

Introduction and Motivation

This paper introduces the concept of D-coloring, a new variant in the hierarchy of edge colorings situated between classical proper edge coloring, B-coloring (rainbow $C_4$), and the strong chromatic index. Specifically, a D-coloring of a graph is a proper edge coloring such that every diamond subgraph (i.e., $K_4$ minus one edge) is rainbow. The D-chromatic index $\chi'_D(G)$ is defined as the minimal number of colors required for such a coloring. The motivation stems from developments around B-coloring, as articulated in work by Gyárfás and Sárközy, and the study of strong edge colorings, but D-coloring features strictly weaker constraints than B-coloring while still capturing essential combinatorial structure.

Relation to Prior Work

Strong edge coloring requires adjacent and (distance-two) edges to be colored differently, with the strong chromatic index $\chi'_s(G)$ denoting the minimal number of colors required. The Erdős–Nešetřil conjecture provides asymptotically sharp bounds on $\chi'_s(G)$. A/B/C-colorings, as introduced by Gyárfás and Sárközy, interpolate between proper and strong edge coloring, with B-colorings ensuring that all $C_4$ subgraphs are rainbow. Notably, for B-coloring, the tight upper bound is $\Delta^2$ for a graph with maximum degree $\Delta$, realized by complete bipartite graphs such as $K_{\Delta, \Delta}$.

The present work positions D-coloring as a relaxation: every B-coloring is a D-coloring but not vice versa. Thus, the challenge is to determine tight bounds on $\chi'_D(G)$, analyze the structural implications, and draw connections to classical coloring and Ramsey-type problems.

Main Results

Bounds on the D-Chromatic Index

• The author proves the general upper bound $K_4$0 for all finite simple graphs of maximum degree $K_4$1.
• The author conjectures the optimal bound $K_4$2, paralleling the situation for strong and B-chromatic indices and showing sharpness for the complete graph $K_4$3.
• This conjectured bound is established for all graphs with $K_4$4.

Proof Methodology

The proof of the general bound leverages a local edge-conflict counting approach: an edge "sees" another if they are incident or cohabit a diamond, necessitating different colors. By maximizing the number of conflicts per edge under prescribed neighborhood structures, the argument bounds the palette needed for greedy extension. This mirrors techniques from strong edge coloring but exploits looser constraints intrinsic to D-coloring.

For small $K_4$5 ($K_4$6), the work classifies all extremal neighborhood types, accounting for overlaps and intersections among neighborhoods to control the number of edges seen by each edge. The absence of minimal counterexamples for $K_4$7 is demonstrated through explicit combinatorial analysis, exploiting structural constraints of regular graphs and extension lemmas for partial colorings.

Behavior on Planar Graphs

In analogy with known results and conjectures for B-coloring, the author discusses D-chromatic indices for planar graphs, noting that for $K_4$8, the upper bound $K_4$9 must hold. Conjecture is made that the following are sharp for planar graphs:

• $\chi'_D(G)$0 when $\chi'_D(G)$1,
• $\chi'_D(G)$2 when $\chi'_D(G)$3,
• $\chi'_D(G)$4 when $\chi'_D(G)$5.

Sharpness is shown via construction, e.g., for $\chi'_D(G)$6 with an added edge.

Technical Implications

The D-coloring paradigm significantly broadens the class of graphs and subgraph structures for which tight control on edge-color diversity within local subgraphs is possible, while subjecting colorings to weaker constraints than strong edge coloring. This has several theoretical implications:

• The quadratic (in $\chi'_D(G)$7) dependence tightens compared to B-coloring, revealing more nuanced interplay between local and non-local constraints in edge coloring.
• The proof methodology suggests generality for extending classical greedy and local adjustment techniques to edge colorings which enforce rainbow conditions on prescribed subgraph families.
• The relationship between D-coloring and induced Ramsey-type phenomena is apparent, as ensuring rainbowness in all diamonds can be viewed as a forbidden monochromatic subgraph condition.

Potential Practical Consequences

These results provide improved palette size requirements for coloring settings encountered in resource assignment with local incompatibility constraints, especially where conflict is dictated by diamond subgraph structures (e.g., communication networks with specific interference patterns). For planar (i.e., topologically embeddable, sparse) graphs, the bounds suggest that palette requirements for such local conflict-free assignments are significantly subquadratic in $\chi'_D(G)$8. The explicit handling of small-degree cases opens the path for algorithmic approaches for low-degree networks.

Directions for Future Research

Implications and avenues for further investigation include:

• Resolving the conjecture $\chi'_D(G)$9 for $\chi'_s(G)$0, sharpening counting arguments or finding lower bounds via extremal constructions.
• Establishing bounds for D-coloring in other families (e.g., graphs of bounded genus, bipartite graphs, or graphs excluding other fixed subgraphs).
• Algorithmic realization of D-coloring assignments in polynomial time for relevant graph classes.
• Exploring chromatic-structural phenomena: how the spectrum between proper, D-, B-, and strong coloring affects other graph invariants like matching number or independence number.

Conclusion

This paper defines and explores D-coloring as an intermediary between proper and B/strong edge colorings, proving sharp bounds for small degrees and a general quadratic upper bound in terms of maximum degree. The work strengthens the understanding of how local rainbow constraints within prescribed subgraphs impact global edge coloring requirements, outlining strengthened bounds for planar and certain dense graphs, and motivates future study on algorithmic aspects and further tightening of the D-chromatic index.