Subgraphs in Which Each Pair of Edges Lies in a Short Common Cycle
نویسنده
چکیده
By a k-graph we mean a pair G = (V,E) consisting of a finite set V of vertices and a collection E of distinct k-element subsets of V called the edges of G . Our object here is to show that each such k-graph with sufficiently many vertices and sufficiently many edges must contain a subgraph H (that is, a "sub-k-graph" H), also having many edges, and having the property that each pair of edges of H lies together in a common subgraph of G which is a type of k-graph "cycle" . In particular for graphs (k=2) we show that each pair of edges of the subgraph H lies together in a cycle of length 4 or one of length 6 in H in the usual graph-theoretic sense, with any two edges of H which share a common vertex being in a cycle of length 4 . Our definition of a "k-cycle" for k > 2 involves the notion of a "separating edge" which was used by Lovász [6] in the formulation of his definition of a "k-forest" .
منابع مشابه
DESIGNING AN OPTIMIZATION MODEL FOR PREVENTING THE WASTE TIME IN THE ACTIVITY CYCLE OF AN ORGANIZATION
This article proposes an optimization model for preventing the waste of time in the educational and research activities' cycle of an organization such as a university. For this purpose and in order to increase efficiency and prevent the waste of time; the graph theory models have been used. The educational and research activities diagrams of a supposed university is drawn by the use of graphs t...
متن کاملMore Results on Subgraphs with Many Short Cycles
1 . Introduction In a previous paper [2] the senior authors proved the following theorem: Let G, = GI (n ;cn 2 ) be a graph of n vertices and cn 2 edges, c a positive constant . Then for n sufficiently large there is always a subgraph G 2 = G2 (m ;f(c)n 2 ) of G, every two edges of which lie together on a cycle of length at most. 6 in G 2 , and if two edges of G 2 have a common vertex they are ...
متن کاملEdge pair sum labeling of some cycle related graphs
Let G be a (p,q) graph. An injective map f : E(G) → {±1,±2,...,±q} is said to be an edge pair sum labeling if the induced vertex function f*: V (G) → Z - {0} defined by f*(v) = ΣP∈Ev f (e) is one-one where Ev denotes the set of edges in G that are incident with a vertex v and f*(V (G)) is either of the form {±k1,±k2,...,±kp/2} or {±k1,±k2,...,±k(p-1)/2} U {±k(p+1)/2} according as p is even or o...
متن کاملOn a problem of Duke-Erdös-Rödl on cycle-connected subgraphs
In this short note, we prove that for β < 1/5 every graph G with n vertices and n2−β edges contains a subgraph G′ with at least cn2−2β edges such that every pair of edges in G′ lie together on a cycle of length at most 8. Moreover edges in G′ which share a vertex lie together on a cycle of length at most 6. This result is best possible up to the constant factor and settles a conjecture of Duke,...
متن کاملCycle Covering
This paper considers the design of a survivable WDM network based on covering the initial network with sub-networks, which are protected independently from each other. We focus on the case where the optical WDM network is a ring, there are requests between any pair of vertices and the covering is done with small cycles. This problem can be modelled as follows: Find a covering of the edges of a ...
متن کامل