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pro vyhledávání: '"Kainen, Paul C."'
Autor:
Kainen, Paul C.
A book embedding of the complete graph $K_n$ needs $\lceil \frac{n}{2} \rceil$ pages and the page-subgraphs can be chosen to be spanning paths (for $n$ even) and one spanning star for $n$ odd. We show that all page-subgraphs can be chosen to be {\rm
Externí odkaz:
http://arxiv.org/abs/2412.00971
Autor:
Gao, Jeffrey, Kainen, Paul C.
A permutation of the elements of a graph is a {\it construction sequence} if no edge is listed before either of its endpoints. The complexity of such a sequence is investigated by finding the delay in placing the edges, an {\it opportunity cost} for
Externí odkaz:
http://arxiv.org/abs/2412.00212
We show that a cyclic vertex order due to Yu, Shao and Li gives a dispersable book embedding for any bipartite circulant.
Comment: 4 pages, 3 figures
Comment: 4 pages, 3 figures
Externí odkaz:
http://arxiv.org/abs/2404.19200
Autor:
Hammack, Richard H., Kainen, Paul C.
Publikováno v:
Mathematics magazine, vol. 97, no. 1, Feb. 2024
It is shown that Euler's theorem for graphs can be generalized for 2-complexes. Two notions that generalize cycle and Eulerian tour are introduced (``circlet'' and ``Eulerian cover''), and we show that for a strongly-connected, pure 2-complex, the fo
Externí odkaz:
http://arxiv.org/abs/2401.00323
Autor:
Hammack, Richard H., Kainen, Paul C.
For strongly connected, pure $n$-dimensional regular CW-complexes, we show that {\it evenness} (each $(n{-}1)$-cell is contained in an even number of $n$-cells) is equivalent to generalizations of both cycle decomposition and traversability.
Com
Com
Externí odkaz:
http://arxiv.org/abs/2401.00084
Autor:
Hammack, Richard H., Kainen, Paul C.
Publikováno v:
American Math. Monthly 128 (4) (2021) 352-359
Beineke, Harary and Ringel discovered a formula for the minimum genus of a torus in which the $n$-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the hypercube's
Externí odkaz:
http://arxiv.org/abs/2401.00070
Autor:
Kainen, Paul C., Overbay, Shannon
Publikováno v:
Bulletin of the Institute of Combinatorics and its Applications, vol 99, 2023, pp 52--57
We show that extending an embedding of a graph $\Gamma$ in a surface to an embedding of a Hamiltonian supergraph can be blocked by certain planar subgraphs but, for some subdivisions of $\Gamma$, Hamiltonian extensions must exist.
Comment: 6 pag
Comment: 6 pag
Externí odkaz:
http://arxiv.org/abs/2303.08306
Autor:
Kainen, Paul C.
Sphere-bases are constructed for the $\mathbb{Z}_2$ vector space formed by the $k$-dimensional subcomplexes, of $n$-simplex (or $n$-cube), for which every $(k{-}1)$-face is contained in a positive even number of $k$-cells; addition is symmetric diffe
Externí odkaz:
http://arxiv.org/abs/2303.07947
Autor:
Kainen, Paul C., Vogt, A.
Publikováno v:
{\it Handbook on Neural Information Processing}, Monica Bianchini, Marco Maggini, Lakhmi C. Jain, Eds., Springer, ISRL Vol. 49, 2013, Chap. 6, pp. 183--214
A Bochner integral formula is derived that represents a function in terms of weights and a parametrized family of functions. Comparison is made to pointwise formulations, norm inequalities relating pointwise and Bochner integrals are established, var
Externí odkaz:
http://arxiv.org/abs/2302.13228
Autor:
Kainen, Paul C.
A construction sequence for a graph is a listing of the elements of the graph (the set of vertices and edges) such that each edge follows both its endpoints. The construction number of the graph is the number of such sequences. We determine this numb
Externí odkaz:
http://arxiv.org/abs/2302.13186