A perfect matching in the complete graph on $2k$ vertices is a set of edges such that no two edges have a vertex in common and every vertex is covered exactly once. Two perfect matchings are said to be $t$-intersecting if they have at least $t$ edges in common. The main result in this paper is an extension of the famous Erdős–Ko–Rado (EKR) theorem [4] to 2-intersecting families of perfect matchings for all values of $k$. Specifically, for $k\ge 3$ a set of 2-intersecting perfect matchings in ${K}_{2k}$ of maximum size has $(2k-5)(2k-7)\cdots \left(1\right)$ perfect matchings.

Revised:

Accepted:

Published online:

Keywords: Erdős–Ko–Rado Theorem, Perfect matchings, Association scheme, Ratio bound, Clique, Coclique, Quotient graphs, Character table.

^{1}; Meagher, Karen

^{1}; Shirazi, Mahsa N.

^{1}

@article{ALCO_2021__4_4_575_0, author = {Fallat, Shaun and Meagher, Karen and Shirazi, Mahsa N.}, title = {The {Erd\H{o}s{\textendash}Ko{\textendash}Rado} theorem for 2-intersecting families of perfect matchings}, journal = {Algebraic Combinatorics}, pages = {575--598}, publisher = {MathOA foundation}, volume = {4}, number = {4}, year = {2021}, doi = {10.5802/alco.169}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/alco.169/} }

TY - JOUR AU - Fallat, Shaun AU - Meagher, Karen AU - Shirazi, Mahsa N. TI - The Erdős–Ko–Rado theorem for 2-intersecting families of perfect matchings JO - Algebraic Combinatorics PY - 2021 SP - 575 EP - 598 VL - 4 IS - 4 PB - MathOA foundation UR - http://archive.numdam.org/articles/10.5802/alco.169/ DO - 10.5802/alco.169 LA - en ID - ALCO_2021__4_4_575_0 ER -

%0 Journal Article %A Fallat, Shaun %A Meagher, Karen %A Shirazi, Mahsa N. %T The Erdős–Ko–Rado theorem for 2-intersecting families of perfect matchings %J Algebraic Combinatorics %D 2021 %P 575-598 %V 4 %N 4 %I MathOA foundation %U http://archive.numdam.org/articles/10.5802/alco.169/ %R 10.5802/alco.169 %G en %F ALCO_2021__4_4_575_0

Fallat, Shaun; Meagher, Karen; Shirazi, Mahsa N. The Erdős–Ko–Rado theorem for 2-intersecting families of perfect matchings. Algebraic Combinatorics, Volume 4 (2021) no. 4, pp. 575-598. doi : 10.5802/alco.169. http://archive.numdam.org/articles/10.5802/alco.169/

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