/library/oar/handle/123456789/405 2026-06-11T03:15:59Z /library/oar/handle/123456789/146228 Title: Isolation of connected graphs Authors: Borg, Peter Abstract: For a connected n-vertex graph G and a positive integer k, let ιk(G) denote the size of a smallest set D of vertices of G such that the graph obtained from G by deleting the closed neighbourhood of D contains no connected graph that has at least k edges. By a result of Caro and Hansberg, ι1(G) ≤ n/3 if n ̸= 2 and G is not a 5-cycle. Let r be the number of vertices of G that have only one neighbour. We show that ι2(G) ≤ (4n−r)/14 if G is not a copy of one of six graphs. We also show that ι3(G) ≤ n/4 if G is neither a triangle nor a 7-cycle. The bounds are sharp. The two new results imply recent results on isolation of graphs. The bound on ι3(G) strengthens the author’s solution to a problem of Caro and Hansberg on isolation of cycles. 2023-01-01T00:00:00Z /library/oar/handle/123456789/143413 Title: Saved by the rook : a case of matchings and Hamiltonian cycles Authors: Abreu, Marién; Gauci, John Baptist; Zerafa, Jean Paul Abstract: The rook graph is a graph whose edges represent all the possible legal moves of the rook chess piece on a chessboard. The problem we consider is the following. Given any set M containing pairs of cells such that each cell of the m1×m2 chessboard is in exactly one pair, we determine the values of the positive integers m1 and m2 for which it is possible to construct a closed tour of all the cells of the chessboard which uses all the pairs of cells in M and some edges of the rook graph. This is an alternative formulation of a graph-theoretical problem presented in [1] involving the Cartesian product G of two complete graphs Km1 and Km2 , which is, in fact, isomorphic to the m1×m2 rook graph. The problem revolves around determining the values of the parameters m1 and m2 that would allow any perfect matching of the complete graph on the same vertex set of G to be extended to a Hamiltonian cycle by using only edges in G. 2025-01-01T00:00:00Z /library/oar/handle/123456789/142588 Title: Analytical solutions to the Laplace equation on a hemispherical domain Authors: Sebu, Cristiana; Amaira, Andrei; Pidcock, Michael Abstract: In this paper, we derive analytical solutions to the Laplace equation in a hemispherical domain subject to two different idealized Neumann boundary conditions. The solutions are given as infinite series, and their convergence is analysed. The theoretical results have been validated by comparing them with numerical results obtained using EIDORS. 2025-01-01T00:00:00Z /library/oar/handle/123456789/142586 Title: Research biography of Jan Boman : mathematician and explorer Authors: Hasanov, Alemdar; Kurasov, Pavel; Novikov, Roman; Quinto, Eric Todd; Sebu, Cristiana; Öktem, Ozan Abstract: This article provides an overview of Jan Boman’s illustrious seventy year career as an approximation theorist, microlocal analyst, and integral geometer. We will include his main mathematical themes and some personal observations. 2025-01-01T00:00:00Z