Contributions to Discrete Mathematics (E-Journal, University of Calgary)
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The Involutive Double Coset Property for String C-groups of Affine Type
Full text linkIn 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 and , we complete the classification by showing it also holds for type and the types for all . As this property is inherited by all string -groups of these types, it follows that the corresponding abstract regular polytopes will have polyhedral realization cones
Linear arboricity of the tensor products of complete multipartite graphs
Full text linkThe linear arboricity of a graph denoted by is the minimum number of linear forests which partition the edge set of Akiyama et al. conjectured that for any -regular graph This conjecture is proved to be true for Also, in [P. Paulraja and S. Sivasankar, Linear Arboricity of the Tensor Products of Graphs, Utilitas Math. 99 (2016) 295--317], the conjecture is proved for the tensor product of complete graphs. Although the conjecture was not proved in general, we have proved that the tensor product of two regular complete multipartite graphs confirms the conjecture in the affirmative
Cayley graphs of order are hamiltonian
Full text linkWe give a computer-assisted proof that if is a finite group of order , where and are distinct primes, then every connected Cayley graph on has a hamiltonian cycle
Multicolor compositions and conjugation of N-color compositions
Full text linkWe introduce -multicolor compositions which generalize the -color compositions that Agarwal first defined twenty-two years ago.
A summand may bear a set of many integer-valued colors which it bounds from above.
We enumerate such multicolored compositions using generating functions. Then we isolate the conjugable class of multicolored compositions known as cracked compositions. A concise exposition of the conjugation of -color compositions is presented in the classical tradition of Percy Alexander MacMahon (1854 - 1929). Our set of conjugable -color compositions turn out to be considerably larger than previously known ones
A linear time approach to three-dimensional reconstruction by discrete tomography
Full text linkThe goal of discrete tomography is to reconstruct an unknown function via a given set of line sums. In addition to requiring accurate reconstructions, it is favourable to be able to perform the task in a timely manner. This is complicated by the presence of ghosts, which allow many solutions to exist in general. In this paper we consider the case of a function where is a finite grid in . Previous work has shown that in the two-dimensional case it is possible to determine all solutions in parameterized form in linear time (with respect to the number of directions and the grid size) regardless of whether the solution is unique. In this work, we show that a similar linear method exists in three dimensions under the condition of nonproportionality.
We show that the condition of nonproportionality is fulfilled in the case of three-dimensional boundary ghosts
On the blocking numbers of some special convex bodies
Full text linkIn this paper, we study the blocking numbers of some special convex bodies. We determine the exact blocking number of a rhombic dodecahedra and a -dimensional cylinder whose base is a -dimensional convex domain. We also estimate that the blocking number of the unit ball in is at most , for 1\leq p<+\infty. In high dimensions, the blocking number of the unit ball in is at most , for \log_{2}d<p<+\infty
Mimimal graphs for completely independent spanning trees and completely independent spanning trees in complete t-partite graph
No full textLet be spanning trees of a graph . For any two vertices of , if the paths from to in these trees are pairwise openly disjoint, then we say that are completely independent spanning trees. In this paper, we give the definition of Minimal graph for completely independent spanning trees and we characterized all Minimal graphs for completely independent spanning trees. Finally, we obtain the number of completely independent spanning trees in complete -partite graph , which is generalize the known result
Construction of the projective plane from the unitary group
Full text linkIn 2013, the first and the second author of this paper described a construction of the projective plane from the unitary group , for . The construction is obtained by using a computer. In the same paper, it is conjectured that in a similar way one can construct the projective plane from the unitary group , for every prime power .
In this paper, we give a construction of a Desarguesian projective plane from a unitary group that confirms this conjecture
The interlacing properties of generalized Narayana polynomials
Full text linkIn this paper, we obtain two new interlacing properties on the zeros of a type of generalized Narayana polynomials arising in the study of the infinite log-concavity of the BorosMoll polynomials. Our tools include a criterion established by Liu and Wang and two new recurrence relations for these Narayana polynomials. The new recurrences are verified with the help of package given by Hou.
Our results also imply the real-rootedness of these Narayana polynomials
Noncrystallographic tail-triangle C-groups of rank 4 and interlacing number 2
Full text linkThis work applies the modular reduction technique to the Coxeter group of rank 4 having a star diagram with labels 5, 3, and or . As moduli, we use the primes in the quadratic integer ring , where , the golden ratio. We prove that each reduced group is a C-group, regardless of the prime used in the reduction. We also classify each reduced group as a reflection group over a finite field, whenever applicable