Distance-Locally Disconnected Graphs

• Autorzy: Mirka Miller , Joe Ryan , Zdeněk Ryjáček
• Języki publikacji: EN

Abstrakty

EN

For an integer k ≥ 1, we say that a (finite simple undirected) graph G is k-distance-locally disconnected, or simply k-locally disconnected if, for any x ∈ V (G), the set of vertices at distance at least 1 and at most k from x induces in G a disconnected graph. In this paper we study the asymptotic behavior of the number of edges of a k-locally disconnected graph on n vertices. For general graphs, we show that this number is Θ(n2) for any fixed value of k and, in the special case of regular graphs, we show that this asymptotic rate of growth cannot be achieved. For regular graphs, we give a general upper bound and we show its asymptotic sharpness for some values of k. We also discuss some connections with cages.

Słowa kluczowe

EN

neighborhood; distance; locally disconnected; cage

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2013
• Tom: 33
• Numer: 1
• Strony: 203-215

Daty

wydano

2013-03-01

online

2013-04-13

Twórcy

autor

Mirka Miller

• University of Newcastle, Australia University of West Bohemia, Pilsen, Czech Republic King’s College, London, UK ITB Bandung, Indonesia

autor

Joe Ryan

• University of Newcastle, Australia

autor

Zdeněk Ryjáček

• University of West Bohemia, Pilsen, Czech Republic Institute for Theoretical Computer Science, Pilsen, Czech Republic University of Newcastle, Australia

Bibliografia

• [1] J.A. Bondy, U.S.R. Murty, Graph Theory (Springer, NewYork, 2008). doi:10.1007/978-1-84628-970-5[Crossref]
• [2] G. Exoo and R. Jajcay, Dynamic cage survey, Electron. J. Combin. 18 (2011) #DS16.
• [3] F. Lazebnik, V.A. Ustimenko and A.J. Woldar, New upper bounds on the order of cages, Electron. J. Combin. 4(2) (1977) R13.
• [4] L. Lovász, J. Pelikán and K. Vesztergombi, Discrete Mathematics: Elementary and Beyond (Springer, NewYork, 2003).
• [5] Z. Ryjáček, N2-locally disconnected graphs, Discrete Math. 121 (1993) 189-193. doi:10.1016/0012-365X(93)90551-4[Crossref]
• [6] Z. Ryjáček and B. Zelinka, Locally disconnected graphs with large numbers of edges, Math. Slovaca 37(2) (1987) 195-198.
• [7] B. Zelinka, Two local properties of graphs, ˇ Casop. Pˇest. Mat. 113 (1988) 113-121.

Identyfikatory

DOI

10.7151/dmgt.1657