On the Directed Hamilton-Waterloo Problem with Two Cycle Sizes

The Directed Hamilton-Waterloo Problem asks for a directed $2$-factorization of the complete symmetric digraph $K_v^*$ where there are two non-isomorphic $2$-factors. In the uniform version of the problem, factors consist of either directed $m$-cycles or $n$-cycles. In this paper, necessary conditions for a solution to this problem are given, and the problem is completely solved for the factors with $(m, n)\in \{(4,6),(4,8),(4,12),(4,16),(6,12),(8,16)\}$. Furthermore, the problem is solved for $(m, n)\in \{(3,5),(3,15),(5,15)\}$ when $v$ is odd with a few possible exceptions

On the double covers of a line graph.

Let $L(X)$ be the line graph of graph $X$. Let $X^{\prime\prime}$ be the Kronecker product of $X$ by $K_2$. In this paper, we see that $L(X^{\prime\prime})$ is a double cover of $L(X)$. We define the symmetric edge graph of $X$, denoted as $\rm{ga}(X)$ which is also a double cover of $L(X)$. We study various properties of $\rm{ga}(X)$ in relation to $X$ and the relationship amongst the three double covers of $L(X)$ that are $L(X^{\prime\prime}),\rm{ga}(X)$ and $L(X)^{\prime\prime}$. With the help of these double covers, we show that for any integer $k\geq 5$, there exist two equienergetic graphs of order $2k$ that are not cospectral

A different approach to Gauss Fibonacci polynomials

In this paper with the help of higher order Fibonacci polynomials, we introduce higher order Gauss Fibonacci polynomials that generalize the Gauss Fibonacci polynomials studied by Özkan and Taştan. We give a recurrence relation, Binet-like formula, generating and exponential generating functions, summation formula for the higher order Gauss Fibonacci polynomials. Moreover, we give two special matrices that we call $Q^{(s)}(x)$ and $P^{(s)}(x),$ respectively. From these matrices, we obtain a matrix representation and derive the Cassini\u27s identity of higher order Gauss Fibonacci polynomials

(G, s)-Transitive Graphs Of Valency 15

Let $X$ be a finite simple undirected graph and $G\leq \rm{Aut}(X)$. If $G$ is transitive on the set of $s$-arcs but not on the set of $(s+1)$-arcs of $X$, then $X$ is called $(G, s)$-transitive. In this paper, we determine the structure of the vertex-stabilizer $G_{v}$ when $X$ is a connected $(G,s)$-transitive graph of valency $15$. We also give some examples to show that each type of $G_{v}$ with $s\ge 2$, can be realized

The existence of hypersolids in Ed\mathbb{E}^d whose Heesch number is d−1d-1

In a recent article, it was shown that the Heesch number in $\mathbb{E}^d$ is asymptotically unbounded for $d\to\infty$, by showing that, for each $d$ of the form $2^k$, there exists a hypersolid in $\mathbb{E}^d$ whose Heesch number equals $d-1$. We here show that the same holds not only for $d$ of the form $2^k$, but for any $d$, $d\geqslant 2$

The Involutive Double Coset Property for String C-groups of Affine Type

In this article, we complete the classification of infinite affine Coxeter group types with the property that every double coset relative to the first parabolic subgroup is represented by an involution. This involutive double coset property was established earlier for the Coxeter groups of type $\tilde{C}_2$ and $\tilde{G}_2$, we complete the classification by showing it also holds for type $\tilde{F}_4$ and the types $\tilde{C}_n$ for all $n$. As this property is inherited by all string $C$-groups of these types, it follows that the corresponding abstract regular polytopes will have polyhedral realization cones

On Freiman-Lev conjecture

Let $A$ be a set of $k$ integers such that $A\subseteq [0, l]$, $0, l\in A$ and $\gcd(A)=1$. Let $2^{\wedge}A$ denote the set of all sums of two distinct elements of $A$. Write $W=\{w\in [0, l]\backslash A: w,w+l\not\in 2^{\wedge}A\}$. In this paper, we obtain the upper bound of $|W|$ with some restrictions on $l$. As an application, we show that the Freiman-Lev conjecture is true for $l=2k-4$ using the structure of $A$ with $|W|=2$

Congruences modulo powers of 3 for 6-colored generalized Frobenius partitions

In his 1984 AMS Memoir, Andrews introduced the family of functions $c\phi_k(n)$, which denotes the number of $k$-colored generalized Frobenius partitions of $n$. In this paper, we prove three congruences and three internal congruences modulo powers of 3 for $c\phi_6(n)$ by utilizing the generating function of $c\phi_6(3n+1)$ due to Hirschhorn. Finally, we conjecture two families of congruences and two families of internal congruences modulo arbitrary powers of 3 for $c\phi_6(n)$, which strengthen a conjecture due to Gu, Wang and Xia in 2016

Fair partitions of the plane into incongruent pentagons

Motivated by a question of R. Nandakumar, we show that the Euclidean plane can be dissected into mutually incongruent convex pentagons of the same area and the same perimeter

Two families of strongly walk regular graphs from three-weight codes over Z4\mathbb{Z_4}

A necessary condition for a $\mathbb{Z_4}$-code to be a three-weight code for the Lee weight is given. Two special constructions of three-weight codes over $\mathbb{Z_4}$ are derived. The coset graphs of their duals are shown to be strongly 3-walk-regular, a generalization of strongly regular graphs