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 Contributions to Discrete MathematicsNumber of Followers: 2     Open Access journal ISSN (Online) 1715-0868 Published by U of Calgary  [18 journals]
• New parity results of sums of partitions and squares in arithmetic
progressions

• Authors: Weiding Hu, Olivia Yao, Taoyan zhao
Abstract: Recently, Ballantine and Merca proved that if $(a,b) \in \{(6,8),\ (8,12),\ (12,24),\ (15,40),\\ (16,48),\ (20,120),\ (21,168)\}$, then $\sum_{ak+1 \ {\rm square}}p(n-k)\equiv 1\ ({\rm mod}\ 2)$ if and only if $bn+1$ is a square. In this paper, we investigate
septuple $(a_1,a_2,a_3,a_4,a_5,a_6,a_7)\in \mathbb{N}^5\times \mathbb{Q}^2$ for which $\sum_{a_1k+a_2 \ {\rm square}}p(a_3a_4^\alpha n+a_6 a_4^\alpha+a_7-k) \equiv 1\ ({\rm mod}\ 2)$ if and only if $a_5n+1$ is a square.
We prove some new parity results of sums of partitions and squares in arithmetic progressions which are analogous to the results due to Ballantine and Merca.
PubDate: 2019-12-26
DOI: 10.11575/cdm.v14i1.62644
Issue No: Vol. 14, No. 1 (2019)

• Feedback vertex number of Sierpi\'{n}ski-type graphs

• Authors: Lili Yuan, Baoyindureng Wu, Biao Zhao
Abstract: The feedback vertex number $\tau(G)$ of a graph $G$ is the minimum number of vertices that can be deleted from $G$ such that the resultant graph does not contain a cycle. We show that $\tau(S_p^n)=p^{n-1}(p-2)$ for the Sierpi\'{n}ski graph $S_p^n$ with $p\geq 2$ and $n\geq 1$. The generalized Sierpi\'{n}ski triangle graph $\hat{S_p^n}$ is obtained by contracting all non-clique edges from the Sierpi\'{n}ski graph $S_p^{n+1}$. We prove that $\tau(\hat{S}_3^n)=\frac {3^n+1} 2=\frac{ V(\hat{S}_3^n) } 3$, and give an upper bound for $\tau(\hat{S}_p^n)$ for the case when $p\geq 4$.
PubDate: 2019-12-26
DOI: 10.11575/cdm.v14i1.62415
Issue No: Vol. 14, No. 1 (2019)

• Anchored Hyperspaces and Multigraphs

• Authors: Gerardo Reyna, Jesús Romero, Ivan Espinobarros
Abstract: Consider a multigraph $X$ as a metric space and $p \in X$. The anchored hyperspace at $p$ is the set  $C_p(X) =$ {$A \subseteq X : p \in A, A$ connected and compact}. In this paper we will prove that $C_p(X)$ is a polytope if in this set is considered the Hausdorff's metric $H$. Further we will show that, if $X$ is a locally connected compact metric space such that $C_p(X)$ is a polytope for each $p \in X$, then $X$ must be a multigraph.

PubDate: 2019-12-26
DOI: 10.11575/cdm.v14i1.62646
Issue No: Vol. 14, No. 1 (2019)

• Sun toughness and $P_{\geq3}$-factors in graphs

• Authors: Sizhong Zhou
Abstract: A $P_{\geq n}$-factor means a path factor with each component having at least $n$ vertices,
where $n\geq2$ is an integer. A graph $G$ is called a $P_{\geq n}$-factor deleted graph if $G-e$
admits a $P_{\geq n}$-factor for any $e\in E(G)$. A graph $G$ is called a $P_{\geq n}$-factor
covered graph if $G$ admits a $P_{\geq n}$-factor containing $e$ for each $e\in E(G)$. In this
paper, we first introduce a new parameter, i.e., sun toughness, which is denoted by $s(G)$. $s(G)$
is defined as follows:
$$s(G)=\min\{\frac{ X }{sun(G-X)}: X\subseteq V(G), \ sun(G-X)\geq2\}$$
if $G$ is not a complete graph, and $s(G)=+\infty$ if $G$ is a complete graph, where $sun(G-X)$
denotes the number of sun components of $G-X$. Then we obtain two sun toughness conditions for a
graph to be a $P_{\geq n}$-factor deleted graph or a $P_{\geq n}$-factor covered graph. Furthermore,
it is shown that our results are sharp.
PubDate: 2019-12-26
DOI: 10.11575/cdm.v14i1.62676
Issue No: Vol. 14, No. 1 (2019)

• Distinguishing number and distinguishing index of neighbourhood corona of
two graphs

• Authors: Saeid Alikhani, Samaneh Soltani
Abstract: The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. The neighbourhood corona of two graphs $G_1$ and $G_2$ is denoted by $G_1 \star G_2$ and is the graph obtained by taking one copy of $G_1$ and $V(G_1)$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. In this paper we describe the automorphisms of the graph $G_1\star G_2$. Using results on automorphisms, we study the distinguishing number and the distinguishing index of $G_1\star G_2$. We obtain upper bounds for $D(G_1\star G_2)$ and $D'(G_1\star G_2)$.
PubDate: 2019-12-26
DOI: 10.11575/cdm.v14i1.62665
Issue No: Vol. 14, No. 1 (2019)

• On the illumination of a class of convex bodies

• Authors: Senlin Wu, Ying Zhou
Abstract: We study Boltyanski’s illumination problem (or Hadwiger's covering problem) for the class of convex bodies in $\mathbb{R}^n$ consisting of convex hulls of a pair of compact convex sets contained in two parallel hyperplanes of $\mathbb{R}^n$. This special case of the problem is completely solved when $n=3$.
PubDate: 2019-12-26
DOI: 10.11575/cdm.v14i1.62685
Issue No: Vol. 14, No. 1 (2019)

• On the classification of automorphisms of trees

• Authors: Samuel Coskey, Kyle Beserra
Abstract: We identify the complexity of the classification problem for automorphisms of a given countable regularly branching tree up to conjugacy. We consider both the rooted and unrooted cases. Additionally, we calculate the complexity of the conjugacy problem in the case of automorphisms of several nonregularly branching trees.
PubDate: 2019-12-26
DOI: 10.11575/cdm.v14i1.62638
Issue No: Vol. 14, No. 1 (2019)

• Proof of a conjecture of Z.W. Sun

• Authors: Olivia Yao, Yan Zhang
Abstract: Rec ently, Sun defined a newsequence $a(n)= \sum_{k=0}^n {n\choose 2k}{2k\choose k}\frac{1}{2k-1}$, which can be viewed as an analogue of Motzkin numbers. Sun conjectured that the sequence $\{\frac{a(n+1)}{a(n)}\}_{n\geq 5}$ is strictly increasing with limit 3, and the sequence
$\{ \sqrt[n+1]{a(n+1)}/\sqrt[n]{a(n)}\}_{n\geq 9}$ is strictly decreasing with limit 1. In this paper, we confirm Sun's conjecture.
PubDate: 2019-12-26
DOI: 10.11575/cdm.v14i1.62700
Issue No: Vol. 14, No. 1 (2019)

• Generating Special Arithmetic Functions by Lambert Series Factorizations

• Authors: Maxie Dion Schmidt, Mircea Merca
Abstract: We summarize the known useful and interesting results and formulas we have discovered so far in this collaborative article summarizing  results from two related articles by Merca and Schmidt arriving at related so-termed Lambert series factorization theorems. We unify the matrix representations that underlie two of our separate papers, and which commonly arise in identities involving partition functions and other functions generated by Lambert series. We provide a number of properties and conjectures related to the inverse matrix entries defined in Schmidt's article and the Euler partition function $p(n)$ which we prove through our new results unifying the expansions of the Lambert series factorization theorems within this article.
PubDate: 2019-12-25
DOI: 10.11575/cdm.v14i1.62425
Issue No: Vol. 14, No. 1 (2019)

• Eulerian polynomials and polynomial congruences

• Authors: Masahiko Yoshinaga, Kazuki Iijima, Kyouhei Sasaki, Yuuki Takahashi
Abstract: We prove that the Eulerian polynomial satisfies certain polynomial congruences. Furthermore, these congruences characterize the Eulerian polynomial.
PubDate: 2019-12-25
DOI: 10.11575/cdm.v14i1.62485
Issue No: Vol. 14, No. 1 (2019)

• On 2-adic behavior of the number of domino tilings on torus

• Authors: Weidong Cheng, Xuejun Guo
Abstract: We study the 2-adic behavior of the number of domino tilings of a $2(2n+1)\times 2(2n+1)$ torus. We show that this number is of the form $2^{4n+3}g(n)^2+2^{8n+2}(2n+1)^{4n}h(n)$, where $g(n)$ and $h(n)$ are odd positive integers. Moreover, we prove that $g(n)$ and $h(n)$ are uniformly continuous under the 2-adic metric and invariant under interchanging $n$ and $-1-n$. This paper is an analogy of Henry Cohn's results for $2n\times 2n$ squares (Electron. J. Combin. 6 (1999)).
PubDate: 2019-12-25
DOI: 10.11575/cdm.v14i1.62673
Issue No: Vol. 14, No. 1 (2019)

• Hamiltonian-connectedness of triangulations with few separating triangles

• Authors: Nico Van Cleemput
Abstract: We prove that 3-connected plane triangulations containing a single edge contained in all separating triangles are hamiltonian-connected. As a direct corollary we have that 3-connected plane triangulations with at most one separating triangle are hamiltonian-connected. In order to show bounds on the strongest form of this theorem, we proved that for any s >= 4 there are 3-connected triangulation with s separating triangles that are not hamiltonian-connected. We also present computational results which show that all `small' 3-connected triangulations with at most 3 separating triangles are hamiltonian-connected.
PubDate: 2019-12-25
DOI: 10.11575/cdm.v14i1.62582
Issue No: Vol. 14, No. 1 (2019)

• A wide class of Combinatorial matrices related with Reciprocal Pascal and
Super Catalan matrices

• Authors: Emrah Kilic, Helmut Prodinger
Abstract: In this paper, we present a number of combinatorial matrices that are generalizations or variants of the super Catalan matrix and the reciprocal Pascal matrix. We present explicit formulæ for LU-decompositions of all the matrices and their inverses. To prove the claimed results, we mainly use the celebrated Zeilberger algorithm.
PubDate: 2019-12-25
DOI: 10.11575/cdm.v14i1.62631
Issue No: Vol. 14, No. 1 (2019)

• Cyclic Orthogonal Double Covers of Circulants by Certain Nerve Cell
Graphs.

Abstract: In this paper, we present a definition to a nerve cell graph. We construct
the cyclic orthogonal double cover of the circulant graphs by the disjoint
union of a graph and a nerve cell graph.
PubDate: 2019-12-25
DOI: 10.11575/cdm.v14i1.62428
Issue No: Vol. 14, No. 1 (2019)

• Hamiltonian paths in $m \times n$ projective checkerboards

• Authors: Dallan McCarthy, Dave Witte Morris
Abstract: For any two squares $\iota$ and $\tau$ of an $m \times n$ checkerboard, we determine whether it is possible to move a checker through a route that starts at $\iota$, ends at $\tau$, and visits each square of the board exactly once. Each step of the route moves to an adjacent square, either to the east or to the north, and may step off the edge of the board in a manner corresponding to the usual construction of a projective plane by applying a twist when gluing opposite sides of a rectangle. This generalizes work of M. H. Forbush et al. for the special case where $m = n$.
PubDate: 2019-12-24
DOI: 10.11575/cdm.v14i1.62399
Issue No: Vol. 14, No. 1 (2019)

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