Perfect 2-Colorings of the Grassmann Graph of Planes

Abstract

We construct an infinite family of intriguing sets, or equivalently perfect 2-colorings, that are not tight in the Grassmann graph of planes of PG$(n,q)$, $n\ge 5$ odd, and show that the members of the family are the smallest possible examples if $n\ge 9$ or $q\ge 25$.

Completely regular codes in Johnson and Grassmann graphs with small covering radii

“…We show that the code of (k + 1)-subspaces not containing subspaces of (k − 1) − (n, k, 1) q -design is completely regular in the Grassmann graph of (k + 1)-subspaces. This generalizes a series of De Winter and Metsch [25] and [1]. In Sections 5 and 6 we go further by applying this idea to the most symmetric designs, i.e.…”

“…In this case a binary linear programming solver found solutions of system (6) for 8 different values of γ 1 . Two of the codes (with γ 1 = 9 and 21) were previously obtained in [25].…”

“…During the years, researchers used several different names for the completely regular codes with covering radius 1. These objects could be equivalently defined in terms of: equitable 2-partitions [28], [30], perfect 2-colorings [13], [1], [26], [19], [25], intriguing sets [11] and others.…”

“…The completely regular codes in the Johnson graphs having strength 1 were characterized in the following cases: the minimum distance at least 3 by Martin in [22] and covering radius 1 recently by Vorob'ev [30]. De Winter and Metsch in [25] considered completely regular codes with covering radius 1 and strength 1 in the Grassmann graphs of diameter 3. They found two new series of examples of such codes: the 3-subspaces that do not contain a space of 2-spread and the code arising from symplectic polar space.…”

Perfect 2‐colorings of Hamming graphs

“…A very nice result of Potapov [41] shows a one‐to‐one correspondence between the perfect 2‐colorings of the Hamming graph and the Boolean‐valued functions on attaining a bound that connects the correlation immunity of the function, the density of ones, and the average 0–1‐contact number (the number of neighbors with function value 1 for a given vertex with value 0). Besides the Hamming graphs, distance‐regular graphs where perfect colorings have been studied include Johnson graphs, see, for example, [1,16], Latin‐square graphs [2], halved hypercubes [32], Grassmann graphs , see, for example, [10,37]. In finite geometry, perfect 2‐colorings are studied as intriguing sets, see, for example, [3].…”