Contributions to Discrete Mathematics (E-Journal, University of Calgary)
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On the enumeration of a class of toroidal graphs
Get PDFWe present enumerations of a class of toroidal graphs which are called semi-equivelar maps. Semi-equivelar maps are generalizations of equivelar maps. There are eight non-isomorphic types of semi-equivelar maps on the torus: , , , , , , , . We attempt to classify all these maps
String C-groups of order 1024
Get PDFThis paper determines the non-degenerate string C-groups of order1024. For groups of rank 3, we use the technique of central extension ofstring C-groups of order 512. For groups of rank at least 4, we compute forquotients of universal string C-groups
Some remarks on the lonely runner conjecture
Get PDFThe lonely runner conjecture of Wills and Cusick, in its most popular formulation, asserts that if runners with distinct constant speeds run around a unit circle starting at a common time and place, then each runner will at some time be separated by a distance of at least 1/(n+1) from the others. In this paper we make some remarks on this conjecture. Firstly, we can improve the trivial lower bound of 1/(2n) slightly for large n, to (1/(2n)) + (c \log n)/(n^2 (\log\log n)^2) for some absolute constant c>0; previous improvements were roughly of the form (1/(2n)) + c/n^2. Secondly, we show that to verify the conjecture, it suffices to do so under the assumption that the speeds are integers of size n^{O(n^2)}. We also obtain some results in the case when all the velocities are integers of size O(n)
Decomposition of the complete bipartite graph with a 1-factor removed into paths and stars
Get PDFLet P_k denote a path on k vertices, and let S_k denote a star with k edges. For graphs F, G, and H, a decomposition of F is a set of edge-disjoint subgraphs of F whose union is F. A (G,H)-decomposition of F is a decomposition of F into copies of G and H using at least one of each. In this paper, necessary and sufficient conditions for the existence of the (P_{k+1},S_k)-decomposition of the complete bipartite graph with a 1-factor removed are given
Binding number, minimum degree and (g,f)-factors of graphs
Get PDFLet a and b be integers with 2<= a< b, and let G be a graph of order n with n>= (a+b-1)^2/(a+1) and the minimum degree \delta(G)<= 1+(((b-2)n)/(a+b-1)).Let g and f be nonnegative integer-valued functions defined on V(G) such that a<= g(x)<f(x)<= b for each x in V(G).We prove that if the binding number bind(G)>=1+((b-2)/(a+1)), then G has a (g,f)-factor
Face Module for Realizable Z-matroids
Get PDFIn this work, we define the face ring for a matroid over Z. Its Hilbert series is, indeed, the expected specialization of the Grothendieck-Tutte polynomial defined by Fink and Moci in [10]
A lower bound on the hypergraph Ramsey number R(4,5;3)
Get PDFThe finite version of Ramsey\u27s theorem says that for positive integers r, k, a_1,... ,a_r, there exists a least number n=R(a_1, \ldots, a_r; k) so that if X is an n-element set and all k-subsets of X are r-coloured, then there exists an i and an a_i-set A so that all k-subsets of A are coloured with the ith colour.In this paper, the bound R(4, 5; 3) >= 35 is shown by using a SAT solver to construct a red--blue colouring of the triples chosen from a 34-element set
On the combinatorics of modified lattice paths and generalized --series
No full textRecently, Agarwal and Sachdeva, 2017, proved two Rogers- Ramanujan type identities for modified lattice paths by establishing a bijection between split (n + t)-color partitions and the modified lattice paths. In this paper, we interpret four generalized basic series combinatorially in terms of modied lattice paths by using a similar bijection. This leads to four new Rogers{Ramanujan type identities for modified lattice paths
Loose Hamiltonian cycles forced by large (k-2)-degree - sharp version
Get PDFWe prove for all k\geq 4 and 1\leq \ell <k/2 the sharp minimum (k-2)-degree bound for a k-uniform hypergraph H on n vertices to contain a Hamiltonian \ell-cycle if k-\ell divides n and n is sufficiently large. This extends a result of Han and Zhao for 3-uniform hypegraphs
On the order of appearance of product of consecutive Fibonacci numbers
No full textLet be the th Fibonacci number. For each positive integer , the order of appearance of , denoted by , is the smallest positive integer such that divides . Recently, D. Marques has obtained a formula for , , and . In this paper, we extend Marques\u27 result to the case , for every . For instance, we prove that, for , where