Multicolor $K_r$-Tilings with High Discrepancy

Abstract: We study the minimum degree threshold $δ{r,q}$ guaranteeing the existence of $K_r$-tilings of high discrepancy in any $q$-edge-coloring. Balogh, Csaba, Pluhár and Treglown handled the 2-color case, proving that $δ{r,2} = \frac{r}{r+1}$ for all $r \geq 3$. Here we determine $δ{r,q}$ for all $q$ large enough, namely $q \geq \binom{r}{2}$. For example, we show that for $r \geq 4$, $δ{r,q} = \frac{r}{r+1}$ for $\binom{r}{2} \leq q \leq \binom{r+1}{2}$ and $δ{r,q} = \frac{r-1}{r}$ for $q \geq \binom{r+1}{2}+2$. Thus, $δ{r,q}$ has a phase transition at $q = \binom{r+1}{2}$, where it drops from $\frac{r}{r+1}$ and then stabilizes at the existence threshold $\frac{r-1}{r}$. We also show that $δ_{r,q} \leq \frac{r}{r+1}$ for all $r,q$, supplementing and giving a new proof for the result of Balogh, Csaba, Pluhár and Treglown.

• The paper establishes precise degree thresholds ensuring the existence of highly discrepant K_r-tilings under any q-coloring.
• It employs a blend of extremal graph constructions, cleaning procedures, and absorbing techniques to demonstrate sharp phase transitions in discrepancy thresholds.
• The results have practical implications for network design and theoretical extensions in combinatorial coloring and tiling problems.

Summary of "Multicolor $K_r$-Tilings with High Discrepancy" (2603.29277)

Problem Formulation and Background

The paper investigates degree thresholds in graphs that ensure the existence of highly discrepant $K_r$-tilings under arbitrary $q$-edge-colorings. Discrepancy is quantified as the extent by which a particular color monopolizes more edges than expected in a $K_r$-tiling relative to uniform distribution. The minimum degree threshold $\delta_{r,q}$ is defined as the smallest value ensuring, for sufficiently large $n$ (with $r | n$), that any $q$-coloring of $G$ (with $\delta(G) \geq \delta n$) contains a $K_r$0-tiling exhibiting discrepancy at least $K_r$1 for some $K_r$2.

Prior results established $K_r$3 for all $K_r$4 [BCPT:21], and the tight existence threshold for uncolored $K_r$5-tilings at $K_r$6 (Hajnal–Szemerédi theorem). This paper generalizes discrepancy thresholds to the multicolored case and analyzes the behavior of $K_r$7 as a function of $K_r$8.

Strong Results and Structural Phase Transitions

A primary contribution is establishing $K_r$9 for all large $q$0 (specifically, $q$1). For $q$2, the authors show:

• $q$3 for $q$4,
• $q$5 for $q$6,
• $q$7 for $q$8,

highlighting a sharp phase transition at $q$9 where the threshold drops to its minimum possible value, corresponding to the existence threshold.

For $K_r$0, the structure is more nuanced: the threshold drops from $K_r$1 to $K_r$2 at $K_r$3, to $K_r$4 at $K_r$5, and then stabilizes at $K_r$6 for $K_r$7.

For small $K_r$8, a divisibility condition determines whether the upper bound $K_r$9 is tight. When certain arithmetic conditions hold (e.g., $\delta_{r,q}$0 or $\delta_{r,q}$1, given $\delta_{r,q}$2), extremal constructions yield the requisite tightness. The main open problem is characterizing $\delta_{r,q}$3 when $\delta_{r,q}$4 but the divisibility condition fails.

Methodology and Technical Innovations

The argument blends extremal graph theory, combinatorial constructions, and color discrepancy analysis. Lower bounds are achieved through multipartite graph constructions with carefully designed color allocations and partition sizes. These constructions ensure that every $\delta_{r,q}$5-tiling has zero discrepancy under the provided $\delta_{r,q}$6-coloring.

Upper bounds involve a cleaning procedure based on a multicolor analog of the graph removal lemma, absorbing techniques, and structural transfer arguments. Templates—subgraphs exhibiting two $\delta_{r,q}$7-tilings with distinct color profiles—are central to the analysis; their existence enables the transfer of discrepancy via blow-up operations. When templates are rare, the authors extract large monochromatic (or few-colored) vertex subsets using color profile transferal, bowtie lemmas, and clique chains. The analysis is inductive, requiring careful handling of the color distribution across increasingly large cliques and neighborhoods.

Key numerical results:

• For large $\delta_{r,q}$8, the minimum discrepancy fraction per color in some $\delta_{r,q}$9-tiling is at least $n$0 where $n$1.
• The construction demonstrates the rigidity of the threshold via explicit counts showing that every $n$2-tiling cannot exceed the balanced allocation.

Contradictory and Bold Claims

The paper asserts and proves that:

• The upper bound $n$3 holds for all $n$4, even as $n$5 increases, contradicting intuition that higher color diversity should necessitate stronger degree conditions for discrepancy.
• There is a phase transition point with discontinuous drop of the discrepancy threshold, a phenomenon rarely seen in classical extremal graph theory.

Practical and Theoretical Implications

Practically, these results impose tight constraints on network designs (multi-channel communication, resource allocation) where partitioning into well-discrepant highly-connected subgraphs is desired under colored constraints. The phase transition behavior guides the selection of network parameters to guarantee such partitioning.

Theoretically, the modular discrepancy analysis and the template technique generalize to other combinatorial objects (e.g., matchings, Hamilton cycles), connecting coloring discrepancy with tiling existence in dense structures. The phase transition mechanisms could inform hypergraph tiling thresholds and the design of robust absorbing methods in colored settings.

Future developments may address the remaining characterization problem for small $n$6, extend template arguments to more general graphs (e.g., bounded-degree or non-partitionable graphs), and apply discrepancy methods to randomized colorings, stochastic network models, or multicolor Turán-type problems.

Conclusion

This work provides a comprehensive determination of minimum degree thresholds guaranteeing highly discrepant $n$7-tilings in multicolored settings, identifying structural phase transitions and tight extremal bounds. The methodology combines construction, inductive transfer analysis, and discrepancy-focused combinatorics, yielding results of central importance in the theory of colored tilings and their applications to extremal and probabilistic combinatorics.