[Paper Review] Lower bounds for rainbow Tur\'{a}n numbers of paths and other trees

Daniel Johnston, Puck Rombach|arXiv (Cornell University)|Jan 10, 2019

Limits and Structures in Graph Theory15 references4 citations

TL;DR

This paper establishes new lower bounds for the rainbow Turán number $ ex^*(n, F) $, showing that for any path $ P_k $ with $ k \geq 3 $ edges, $ ex^*(n, P_k) \geq \frac{k}{2}n + O(1) $, generalizing prior results. The authors use structured edge-colored graphs—specifically $ D_{2^s}^* $, a subgraph of the complete graph $ K_{2^s}^* $ with colors based on Hamming differences—to construct dense edge-colored graphs without rainbow copies of $ P_k $, and extend the method to brooms and other small-diameter caterpillars.

ABSTRACT

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n, F)$, is the rainbow Tur\\'{a}n number of $F$. We show that $ex^*(n,P_k)\\geq \\frac{k}{2}n + O(1)$ where $P_k$ is a path on $k\\geq 3$ edges, generalizing a result by Maamoun and Meyniel and by Johnston, Palmer and Sarkar. We show similar bounds for brooms on $2^s-1$ edges and diameter $\\leq 10$ and a few other caterpillars of small diameter.

Motivation & Objective

• To generalize the known lower bound $ ex^*(n, P_k) = \frac{k}{2}n + O(1) $ from $ k \in \{3,4\} $ to all $ k \geq 3 $, establishing a broader structural understanding of rainbow Turán extremal graphs.
• To extend the method to other trees beyond paths, particularly brooms and caterpillars of small diameter, by constructing edge-colored graphs that avoid rainbow copies of these trees.
• To investigate the extremal edge density in properly edge-colored graphs that avoid rainbow copies of specific trees, contributing to the broader rainbow Turán problem for bipartite and forest-like graphs.
• To explore the utility of algebraic and combinatorial constructions—specifically $ D_{2^s}^* $, derived from $ \mathbb{F}_2^s $—as tools for generating extremal examples in rainbow Turán theory.

Proposed method

• Construct the edge-colored complete graph $ K_{2^s}^* $ on $ 2^s $ vertices, where each edge $ vw $ is colored by the vector $ v - w \in \mathbb{F}_2^s $, forming a proper edge coloring.
• Define the subgraph $ D_{2^s}^* $ as the spanning subgraph of $ K_{2^s}^* $ containing only edges with Hamming distance $ 1 $ or $ s $ between endpoints, ensuring it is $ (s+1) $-regular and properly edge-colored.
• Use the $ D_{2^s}^* $ construction to derive lower bounds on $ ex^*(n, F) $ by counting rainbow paths and cycles in these graphs, leveraging their symmetry and controlled color distribution.
• Apply counting arguments to estimate the number of rainbow copies of $ P_k $, $ C_k $, and other trees in $ D_{2^s}^* $, showing that for $ k = 2^s - 1 $, the number of rainbow $ C_k $ copies is $ \Omega(k^{\ell-1}n) $ for $ \ell \ll k $.
• Generalize the path construction to brooms $ B_{k,l} $ and caterpillars $ CP_{(s_1,\dots,s_t)} $, showing that $ D_{2^s}^* $-based constructions avoid rainbow copies of these trees when $ k = 2^s - 1 $.
• Use extremal graph theory tools, including average degree arguments and path extension lemmas (e.g., if a vertex is an endpoint of a maximal rainbow path of length $ k $, its degree is at most $ 2k-2 $), to bound the number of rainbow paths and validate extremality.

Experimental results

Research questions

• RQ1 Can the lower bound $ ex^*(n, P_k) \geq \frac{k}{2}n + O(1) $ be extended from $ k = 3,4 $ to all $ k \geq 3 $, and is this bound tight?
• RQ2 Do constructions based on $ D_{2^s}^* $, derived from $ \mathbb{F}_2^s $, yield extremal examples for avoiding rainbow copies of other trees, such as brooms and small-diameter caterpillars?
• RQ3 Is the path $ P_k $ the easiest tree (in terms of minimizing $ ex^*(n, F) $) to avoid in properly edge-colored graphs, or are there trees with smaller rainbow Turán numbers?
• RQ4 What is the maximum number of rainbow copies of $ C_k $ in $ K_{2^s}^* $ when $ k = 2^s - 1 $, and how does this compare to the number in $ D_{2^s}^* $?
• RQ5 Are there other subgraphs of $ K_{2^s}^* $, beyond $ D_{2^s}^* $, that efficiently avoid rainbow copies of specific trees, such as $ t $-spiders or other caterpillars?

Key findings

• The paper establishes a general lower bound $ ex^*(n, P_k) \geq \frac{k}{2}n + O(1) $ for all $ k \geq 3 $, extending previous results that were only known for $ k = 3,4 $.
• For brooms $ B_{k,l} $ with $ k = 2^s - 1 $ edges and diameter at most 10, the construction $ D_{2^s}^* $ yields $ ex^*(n, B_{k,l}) \geq \frac{k}{2}n + O(1) $, showing similar extremal behavior to paths.
• For caterpillars $ CP_{(s_1,\dots,s_t)} $ with small diameter and $ k = 2^s - 1 $ edges, the same construction gives $ ex^*(n, F) \geq \frac{k}{2}n + O(1) $, indicating that such trees are also avoidable in dense edge-colored graphs.
• The number of rainbow copies of $ C_k $ in $ D_{2^s}^* $ for $ k = 2^s - 1 $ is at least $ \frac{k(k-1)}{6}n + O(1) $, and more generally $ \Omega(k^{\ell-1}n) $ for $ \ell \ll k $, showing rich rainbow cycle structure.
• The construction $ D_{2^s}^* $ is shown to be $ C_k $-free for $ k = 2^s - 1 $, and thus serves as a valid extremal example for avoiding rainbow $ P_k $, confirming its utility in extremal constructions.
• The paper demonstrates that $ D_{2^s}^* $, a $ (s+1) $-regular graph on $ 2^s $ vertices, avoids rainbow $ P_k $ for $ k = 2^s - 1 $, and its edge density is sufficient to achieve the lower bound $ \frac{k}{2}n + O(1) $.