Contributions to Discrete Mathematics (E-Journal, University of Calgary)
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Anchored Hyperspaces and Multigraphs
Get PDFConsider a multigraph as a metric space and p \in X. The anchored hyperspace at is the set
{A \subseteq X : p \in A, A connected and compact}.
In this paper we will prove that is a polytope if in this set is considered the Hausdorff\u27s metric . Further we will show that, if is a locally connected compact metric space such that is a polytope for each p \in X, then must be a multigraph
New parity results of sums of partitions and squares in arithmetic progressions
Get PDFRecently, Ballantine and Merca proved that if , then if and only if is a square. In this paper, we investigate septuple for which if and only if 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
On 2-adic behavior of the number of domino tilings on torus
No full textWe study the 2-adic behavior of the number of domino tilings of a torus. We show that this number is of the form , where and are odd positive integers. Moreover, we prove that and are uniformly continuous under the 2-adic metric and invariant under interchanging and . This paper is an analogy of Henry Cohn\u27s results for squares (Electron. J. Combin. 6 (1999))
Eulerian polynomials and polynomial congruences
Get PDFWe prove that the Eulerian polynomial satisfies certain polynomial congruences. Furthermore, these congruences characterize the Eulerian polynomial
Hamiltonian paths in projective checkerboards
No full textFor any two squares and of an checkerboard, we determine whether it is possible to move a checker through a route that starts at , ends at , 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
On the classification of automorphisms of trees
Get PDFWe 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
Proof of a conjecture of Z.W. Sun
Get PDFRec ently, Sun defined a newsequence , which can be viewed as an analogue of Motzkin numbers. Sun conjectured that the sequence is strictly increasing with limit 3, and the sequence is strictly decreasing with limit 1. In this paper, we confirm Sun\u27s conjecture
Sun toughness and -factors in graphs
No full textA -factor means a path factor with each component having at least vertices,where is an integer. A graph is called a -factor deleted graph if admits a -factor for any . A graph is called a -factorcovered graph if admits a -factor containing for each . In thispaper, we first introduce a new parameter, i.e., sun toughness, which is denoted by . is defined as follows:if is not a complete graph, and if is a complete graph, where denotes the number of sun components of . Then we obtain two sun toughness conditions for agraph to be a -factor deleted graph or a -factor covered graph. Furthermore,it is shown that our results are sharp
Generating Special Arithmetic Functions by Lambert Series Factorizations
Get PDFWe 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\u27s article and the Euler partition function which we prove through our new results unifying the expansions of the Lambert series factorization theorems within this article
Cyclic Orthogonal Double Covers of Circulants by Certain Nerve Cell Graphs.
No full textIn this paper, we present a definition to a nerve cell graph. We constructthe cyclic orthogonal double cover of the circulant graphs by the disjointunion of a graph and a nerve cell graph