Spanning Subgraph Research Articles

Spanning subgraphs of random graphs

We propose a problem concerning the determination of the threshold function for the edge probability that guarantees, almost surely, the existence of various sparse spanning subgraphs in random graphs. We prove some bounds and demonstrate them in the cases of ad-cube and a two dimensional lattice.

The Evolution of Random Subgraphs of the Cube

Abstract be a random Qn”‐process, that is let Q0 be the empty spanning subgraph of the cube Qn and, for 1 ⩽ t ⩽ M = nN/2 = n2n−1, let the graph Qt be obtained from Qt−1 by the random addition of an edge of Qn not present in Qt−1. When t is about N/2, a typical Qt undergoes a certain “phase transition'': the component structure changes in a sudden and surprising way. Let t = (1 + ϵ) N/2 where ϵ is independent of n. Then all the components of a typical Qt have o(N) vertices if ϵ < 0, while if ϵ > 0 then, as proved by Ajtai, Komlós, and Szemerédi, a typical Qt has a “giant” component with at least α(ϵ)N vertices, where α(ϵ) > 0. In this note we give essentially best possible results concerning the emergence of this giant component close to the time of phase transition. Our results imply that if η > 0 is fixed and t ⩽ (1 − n−η) N/2, then all components of a typical Qt have at most nβ(η) vertices, where β(η) > 0. More importantly, if 60(log n)3/n ⩽ ϵ = ϵn = o(1), then the largest component of a typical Qt has about 2ϵN vertices, while the second largest component has order O(nϵ−2). Loosely put, the evolution of a typical Qn process is such that shortly after time N/2 the appearance of each new edge results in the giant component acquiring 4 new vertices.

Skolem labelled graphs

A Skolem labelled graph is a triple ( G, L, d), where G = (V, E) is a graph and L:V → d, d + 1,…, d+ m satisfying: 1. (a) There are exactly two vertices in V, such that L( v)= d+ i, 0 ⩽ i⩽ m. 2. (b) The distance in G between any two vertices with the same label is the value of the label. 3. (c) If G' is a proper spanning subgraph of G, ( G', L, d) is not a Skolem labelled graph. Note that this definition is different from the Skolem-graceful labelling of Lee, Quach and Wang (1988). When d=1 is not specified it is assumed. We shall establish the following: 1. (1) Any tree can be embedded in a Skolem labelled tree with O( v) vertices. 2. (2) Any graph can be embedded as an induced subgraph in a Skolem labelled graph on O( v 3) vertices. 3. (3) For d=1, we exhibit a Skolem or the minimum hooked Skolem (with as few unabelled vertices as possible) labelling for paths and cycles. 4. (4) For d=1 we exhibit the minimum Skolem labelled gaph containing a path or a cycle of length n as induced subgraph.

On the Structure of Minimum-Weight k-Connected Spanning Networks

The problem of finding a minimum-weight k-connected spanning subgraph of a complete graph, assuming that the edge weights satisfy the triangle inequality, is studied. It is shown that the class of minimum-weight k-edge connected spanning subgraphs can be restricted to those subgraphs which, in addition to the connectivity requirements, satisfy the following two conditions: (I) Every vertex has degree k or $k + 1$; (II) Removing any $1, 2, \cdots ,$ or k edges does not leave the resulting connected components all k-edge connected. For the k-vertex connected case, the parallel result is obtained with “k-edge” replaced by “k-vertex,” with the added technical restriction that $| V |\geqq 2k$ for condition (I) to hold. This generalizes recent work of Monma, Munson, and Pulleyblank for the case $k = 2$.

Existence of Resolvable Path Designs

For k ⩾ 2 the complete multigraph λKn has a factorization into isomorphic spanning subgraphs, each component of which is a path of length k - 1 iff n ≡ 0 (mod k) and λk(n - 1) ≡ 0 (mod 2(k - 1)).

A,b]‐factorization of a graph

AbstractLet a and b be integers such that 0 ⩽ a ⩽ b. Then a graph G is called an [a, b]‐graph if a ⩽ dG(x) ϵ b for every x ϵ V(G), and an [a, b]‐factor of a graph is defined to be its spanning subgraph F such that a ⩽ dF(x) ⩽ b for every vertex x, where dG(x) and dF(x) denote the degrees of x in G and F, respectively. If the edges of a graph can be decomposed into [a.b]‐factors then we say that the graph is [2a, 2a]‐factorable. We prove the following two theorems: (i) a graph G is [2a, 2b)‐factorable if and only if G is a [2am,2bm]‐graph for some integer m, and (ii) every [8m + 2k, 10m + 2k]‐graph is [1,2]‐factorable.

Subcube Coverings of Random Spanning Subgraphs of then-Cube

Let G(n, p) denote the probability space consisting of all spanning subgraphs g of the n-cube En, and the probability is defined as P(g) = pM (1–p) if g contains exactly M edges, 0 < M < n2n–1. Erdos and Spencer investigated the connectedness of such random graphs for fixed probability p, 0 < p < 1 (cf. [1]). In this paper we study coverings of the vertex set of g ϵ G(n, p) by subcubes of En being also subgraphs of g. Note the analogy between such coverings and coverings of the vertex set of spanning subgraphs of the complete graph Kn by cliques.

On the structure of complete graphs without alternating cycles

The structure of edge-coloured complete graphs Kp which do not contain cycles with adjacent lines differently coloured is completely characterized. This characterization further enables us to prove that evey such Kp (p⩾2) has exactly one connected monochromatic spanning subgraph. Moreover, assume that edges in Kp are coloured by k colours so that no vertex is on more than Δ edges of each colour and less than δ edges of each colour. If such a Kp exists then p⩽Δ+1 and p⩾kδ+1, whereas, any one of these inequalities will ensure the existence of such a Kp.