Problem
Research questions and friction points this paper is trying to address.
Ordered Ruzsa-Szemeredi numbers
Matchings of size two
Fully dynamic matching
Extremal graph theory
Exact Ordered Ruzsa-Szemeredi Numbers for Matchings of Size Two
🤖 AI Summary
This study investigates the ordered Ruzsa–Szemerédi numbers for matchings of size two. By establishing a bijection between ordered decompositions and K4-stripping, and employing combinatorial techniques including structural lemmas and isomorph-free backtracking, we exactly determine these values for orders 5 through 19 and narrow the range for order 20, while providing an explicit asymptotic construction. Furthermore, this work reveals irregular failure patterns in the natural equality conjecture, thereby disproving it. These findings not only deepen the understanding of structures within ordered extremal graph theory but also provide precise theoretical foundations for parameterizing update time in dynamic matching algorithms.
📝 Abstract
An ordered Ruzsa-Szemeredi graph is a graph whose edge set is partitioned into equal-size matchings, each induced in the suffix of the ordering that begins with it. Behnezhad and Ghafari introduced them to parametrize the update time of fully dynamic matching, but almost nothing is known about the numbers themselves. Writing f(n) for the largest number of parts when the matchings have size two, we determine f(n) exactly for every order from five to nineteen, narrow order twenty to two consecutive values, and give an explicit asymptotic construction.
The engine is a bijection between ordered decompositions and K_4-peelings of the complete graph, each step deleting a perfect matching from four vertices that currently span a clique. This yields the counting bound floor(n(n-4)/4) at once and reduces equality to whether a cubic or near-cubic remainder is reachable. Structural lemmas cut the candidates to connected bridgeless graphs, and a contraction correspondence carries odd orders to the even census one larger, leaving a finite case analysis that we discharge by isomorphism-free reverse search.
The bound is attained only at orders five through nine and eleven, and missed by exactly one at every other order we reach. Order eleven is thus an isolated exception rather than a parity phenomenon: the natural equality conjecture fails, and fails irregularly. Upper bounds are certified by fail-closed sweeps over complete cubic censuses, and every decomposition is re-checked against the definition by an independent verifier. Which of its two values order twenty takes remains open.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Ordered Ruzsa-Szemeredi Numbers
K4-peelings
Isomorphism-free Reverse Search
Contraction Correspondence
Dynamic Matching
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