Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2009
Volume
Vol. 29
Number
No. 4
Description
Journal Volume
Opuscula Mathematica
Vol. 29 (2009)
Projects
Pages
Articles
Edge condition for hamiltonicity in balanced tripartite graphs
(2009) Adamus, Janusz
A well-known theorem of Entringer and Schmeichel asserts that a balanced bipartite graph of order $2n$ obtained from the complete balanced bipartite $K_{n,n}$ by removing at most $n - 2$ edges, is bipancyclic. We prove an analogous result for balanced tripartite graphs: If $G$ is a balanced tripartite graph of order $3n$ and size at least $3n^{2} - 2n + 2$, then $G$ contains cycles of all lengths.
Cyclability in bipartite graphs
(2009) Amar, Denise; Flandrin, Evelyne; Gancarzewicz, Grzegorz
Let $G = (X,Y,E)$ be a balanced $2$-connected bipartite graph and $S \subset V(G)$. We will say that $S$ is cyclable in $G$ if all vertices of $S$ belong to a common cycle in $G$. We give sufficient degree conditions in a balanced bipartite graph $G$ and a subset $S \subset V(G)$ for the cyclability of the set $S$.
Matrices defined by frames
(2009) Ambroziński, Zbigniew; Rudol, Krzysztof
Matrix representations of bounded Hilbert space operators are considered. The matrices in question are defined with respect to frames, rather than bases. The frames, a generalisation of bases, used extensively in applied harmonic analysis, are overcomplete sequences. We consider some properties related to tight frames, where, up to some multiplicative constant, a form of Parseval Identity takes place. We also describe parts of spectra of operators in terms of their matrices.
On elliptic problems with a nonlinearity depending on the gradient
(2009) Chabrowski, Jan
We investigate the solvability of the Neumann problem $(1.1)$ involving the nonlinearity depending on the gradient. We prove the existence of a solution when the right hand side $f$ of the equation belongs to $L^m(\Omega )$ with $1 \leq m \lt 2$.
α2-labeling of graphs
(2009) Fronček, Dalibor
We show that if a graph $G$ on n edges allows certain special type of rosy labeling (a.k.a. $\rho$-labeling), called $\alpha_2$-labeling, then for any positive integer $k$ the complete graph $K_{2nk+1}$ can be decomposed into copies of $G$. This notion generalizes the $\alpha$-labeling introduced in 1967 by A. Rosa.