Long cycles and paths in distance graphs

نویسندگان

  • Lucia Draque Penso
  • Dieter Rautenbach
  • Jayme Luiz Szwarcfiter
چکیده

For n ∈ N and D ⊆ N, the distance graph P n has vertex set {0, 1, . . . , n− 1} and edge set {ij | 0 ≤ i, j ≤ n − 1, |j − i| ∈ D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs. A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a set D, there is a constant cD such that the greatest common divisor of the integers in D is 1 if and only if for every n, P n has a component of order at least n−cD if and only if for every n, P n has a cycle of order at least n−cD. Furthermore, we discuss some consequences and variants of this result.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Different-Distance Sets in a Graph

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...

متن کامل

Characterization of signed paths and cycles admitting minus dominating function

If G = (V, E, σ) is a finite signed graph, a function f : V → {−1, 0, 1} is a minusdominating function (MDF) of G if f(u) +summation over all vertices v∈N(u) of σ(uv)f(v) ≥ 1 for all u ∈ V . In this paper we characterize signed paths and cycles admitting an MDF.

متن کامل

Powers of cycles, powers of paths, and distance graphs

In 1988, Golumbic and Hammer characterized powers of cycles, relating them to circular-arc graphs. We extend their results and propose several further structural characterizations for both powers of cycles and powers of paths. The characterizations lead to linear-time recognition algorithms of these classes of graphs. Furthermore, as a generalization of powers of cycles, powers of paths, and ev...

متن کامل

Incidence dominating numbers of graphs

In this paper, the concept of incidence domination number of graphs  is introduced and the incidence dominating set and  the incidence domination number  of some particular graphs such as  paths, cycles, wheels, complete graphs and stars are studied.

متن کامل

Relative length of long paths and cycles in graphs with large degree sums

Long Paths and Cycles in Graphs with Large D egree Sums Hikoe Enomoto DEPARTMENT 0 F MATHEMATICS, KElO UNIVERSITY, HlYOSHl3-14-1 KOHOKU-KU, YOKOHAMA KANAGAWA 223, JAPAN Jan van den Heuvel* FACULTY OF APPLIED MATHEMATICS UNIVERSITY OF TWENTE, P.O. BOX217 7500 AE ENSCHEDE THE NETHERLANDS Atsushi Kaneko DEPARTMENT OF ELECTRONIC ENGINEERING KOGAKUIN UNlVERSlm, NISHI-SHINJUKU 1-24-2 SHINJUKU-KU, TOK...

متن کامل

Asteroidal number for some product graphs

The notion of Asteroidal triples was introduced by Lekkerkerker and Boland [6]. D.G.Corneil and others [2], Ekkehard Kohler [3] further investigated asteroidal triples. Walter generalized the concept of asteroidal triples to asteroidal sets [8]. Further study was carried out by Haiko Muller [4]. In this paper we find asteroidal numbers for Direct product of cycles, Direct product of path and cy...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:
  • Discrete Mathematics

دوره 310  شماره 

صفحات  -

تاریخ انتشار 2010