ﻻ يوجد ملخص باللغة العربية
For any positive integers $n, s, t, l$ such that $n geq 10$, $s, t geq 2$, $l geq 1$ and $n geq s+t+l$, a new infinite family of regular 3-hypertopes with type $(2^s, 2^t, 2^l)$ and automorphism group of order $2^n$ is constructed.
A graph is $ell$-reconstructible if it is determined by its multiset of induced subgraphs obtained by deleting $ell$ vertices. We prove that $3$-regular graphs are $2$-reconstructible.
We show the existence of regular combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise permutations.
The 1-2-3 Conjecture, posed by Karo{n}ski, {L}uczak and Thomason, asked whether every connected graph $G$ different from $K_2$ can be 3-edge-weighted so that every two adjacent vertices of $G$ get distinct sums of incident weights. The 1-2 Conjecture
In this paper, we classify regular polytopes with automorphism groups of order $2^n$ and Schlafli types ${4, 2^{n-3}}, {4, 2^{n-4}}$ and ${4, 2^{n-5}}$ for $n geq 10$, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [P
A ${00,01,10,11}$-valued function on the vertices of the $n$-cube is called a $t$-resilient $(n,2)$-function if it has the same number of $00$s, $01$s, $10$s and $11$s among the vertices of every subcube of dimension $t$. The Friedman and Fon-Der-Fla