Degree Sequence of Tight Distance Graphs

A graph G on n vertices is a tight distance graph if there exists a set $D\subseteq \{1,2,\dots ,n-1\}$ such that $V(G)=\{0,1,\dots ,n-1\}$ and $ij\in E(G)$ if and only if $|i-j|\in D$. A characterization of the degree sequences of tight distance graphs is given. This characterization yields a fast method for recognizing and realizing degree sequences of tight distance graphs.     

On the existence of retransmission permutation arrays

We investigate retransmission permutation arrays (RPAs) that are motivated by applications in overlapping channel transmissions. An RPA is an $n\times n$ array in which each row is a permutation of $\{1,\dots ,n\}$, and for $1?i?n$, all $n$ symbols occur in each $i\times ?\frac{n}{i}?$ rectangle in specified corners of the array. The array has types 1, 2, 3 and 4 if the stated property holds in the top left, top right, bottom left and bottom right corners, respectively. It is called latin if it is a latin square. We show that for all positive integers $n$, there exists a type-1, 2, 3, 4 $\mathrm{RPA}\left(n\right)$ and a type-1, 2 latin $\mathrm{RPA}\left(n\right)$.     

Solvable PST-groups,strong Sylow bases and mutually permutable products

A subgroup H of a group G is said to permute with the subgroup K of G if $HK=KH$. Subgroups H and K are mutually permutable (totally permutable) in G if every subgroup of H permutes with K and every subgroup of K permutes with H (if every subgroup of H permutes with every subgroup of K). If H and K are mutually permutable and $H\cap K=1$, then H and K are totally permutable. A subgroup H of G is S-permutable in G if H permutes with every Sylow subgroup of G. A group G is called a PST-group if S-permutability is a transitive relation in G. Let $\{p_1,\dots ,p_n,p_{n+1},\dots ,p_k\}$ be the set of prime divisors of the order of a finite group G with $\{p_1,\dots ,p_n\}$ the set of prime divisors of the order of the normal subgroup N of G. A set of Sylow subgroups $\{P_1,\dots ,P_n,P_{n+1},\dots ,P_k\}$, $P_i\in {\mathrm{Syl}}_{p_i}(G)$, form a strong Sylow system with respect to N if $P_i{P}_j$ is a mutually permutable product for all $i\in \{1,2,\dots ,n\}$ and $j\in \{1,2,\dots ,k\}$. We show that a finite group G is a solvable PST-group if and only if it has a normal subgroup N such that $G/N$ is nilpotent and G has a strong Sylow system with respect to N. It is also shown that G is a solvable PST-group if and only if G has a normal solvable PST-subgroup N and $G/N^{″}$ is a solvable PST-group.     

On tiling the integers with 4-sets of the same gap sequence

Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in various ways has been considered throughout history. Given a set $\{x_1,\dots ,x_n\}$ of integers where $x_1<?<x_n$, let the gap sequence of this set be the unordered multiset $\{d_1,\dots ,d_{n?1}\}=\{x_{i+1}?x_i:i\in \{1,\dots ,n?1\}\}$. This paper addresses the following question, which was explicitly asked by Nakamigawa: can the set of integers be partitioned into sets with the same gap sequence? The question is known to be true for any set where the gap sequence has length at most two. This paper provides evidence that the question is true when the gap sequence has length three. Namely, we prove that given positive integers $p$ and $q$, there is a positive integer $r_0$ such that for all $r\ge r_0$, the set of integers can be partitioned into 4-sets with gap sequence $p,q$, $r$.     

Self-complementary magic squares of doubly even orders

A magic square $M$ in which the entries consist of consecutive integers from $1,2,\dots ,n^2$ is said to be self-complementary of order$n$ if the resulting square obtained from $M$ by replacing each entry $i$ by $n^2+1?i$ is equivalent to $M$ (under rotation or reflection). We present a new construction for self-complementary magic squares of order $n$ for each $n\ge 4$, where $n$ is a multiple of $4$.     

All double generalized Petersen graphs are Hamiltonian

The double generalized Petersen graph $DP(n,t)$, $n\ge 3$ and $t\in {\mathbb{Z}}_n?\left\{0\right\}$, $2\le 2t<n$, has vertex-set $\{x_i,y_i,u_i,v_i\mid i\in {\mathbb{Z}}_n\}$, edge-set $\{\{x_i,x_{i+1}\},\{y_i,y_{i+1}\},\{u_i,v_{i+t}\},\{v_i,u_{i+t}\},\{x_i,u_i\},\{y_i,v_i\}\mid i\in {\mathbb{Z}}_n\}$. These graphs were first defined by Zhou and Feng as examples of vertex-transitive non-Cayley graphs. Then, Kutnar and Petecki considered the structural properties, Hamiltonicity properties, vertex-coloring and edge-coloring of $DP(n,t)$, and conjectured that all $DP(n,t)$ are Hamiltonian. In this paper, we prove this conjecture.     

The coloring complex and cyclic coloring complex of a complete k-uniform hypergraph

In this paper, we study the homology of the coloring complex and the cyclic coloring complex of a complete k-uniform hypergraph. We show that the coloring complex of a complete k-uniform hypergraph is shellable, and we determine the rank of its unique nontrivial homology group in terms of its chromatic polynomial. We also show that the dimension of the $(n?k?1)\text{st}$ homology group of the cyclic coloring complex of a complete k-uniform hypergraph is given by a binomial coefficient. Further, we discuss a complex whose r-faces consist of all ordered set partitions $[B_1,\dots ,B_{r+2}]$ where none of the $B_i$ contain a hyperedge of the complete k-uniform hypergraph H and where $1\in B_1$. It is shown that the dimensions of the homology groups of this complex are given by binomial coefficients. As a consequence, this result gives the dimensions of the multilinear parts of the cyclic homology groups of $\mathbb{C}[x_1,\dots ,x_n]/\{x_{i_1}\dots x_{i_k}|i_1\dots i_k\text{is a hyperedge of}H\}$.     

Balanced diagonals in frequency squares

We say that a diagonal in an array is $\lambda$-balanced if each entry occurs $\lambda$ times. Let $L$ be a frequency square of type $F(n;\lambda )$; that is, an $n\times n$ array in which each entry from $\{1,2,\dots ,m=n∕\lambda \}$ occurs $\lambda$ times per row and $\lambda$ times per column. We show that if $m?3$, $L$ contains a $\lambda$-balanced diagonal, with only one exception up to equivalence when $m=2$. We give partial results for $m?4$ and suggest a generalization of Ryser’s conjecture, that every Latin square of odd order has a transversal. Our method relies on first identifying a small substructure with the frequency square that facilitates the task of locating a balanced diagonal in the entire array.