Fixed Graph Research Articles

Induced Subgraph in Random Regular Graph

Let Gn,d be a random d-regular graph with n vertices, where d = o(n). Given a fixed graph H, YH denotes the number of induced copies of H in Gn,d. In this paper, the authors determine the threshold of the event “YH > 0”, and also obtain the induced subgraph counts inside the threshold interval.

Avoiding small subgraphs in Achlioptas processes

AbstractFor a fixed integer r, consider the following random process. At each round, one is presented with r random edges from the edge set of the complete graph on n vertices, and is asked to choose one of them. The selected edges are collected into a graph, which thus grows at the rate of one edge per round. This is a natural generalization of what is known in the literature as an Achlioptas process (the original version has r = 2), which has been studied by many researchers, mainly in the context of delaying or accelerating the appearance of the giant component.In this article, we investigate the small subgraph problem for Achlioptas processes. That is, given a fixed graph H, we study whether there is an online algorithm that substantially delays or accelerates a typical appearance of H, compared to its threshold of appearance in the random graph G(n, M). It is easy to see that one cannot accelerate the appearance of any fixed graph by more than the constant factor r, so we concentrate on the task of avoiding H. We determine thresholds for the avoidance of all cycles Ct, cliques Kt, and complete bipartite graphs Kt,t, in every Achlioptas process with parameter r ge 2. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

Online balanced graph avoidance games

We introduce and study online balanced coloring games on the random graph process. The game is played by a player we call Painter. Edges of the complete graph with n vertices are introduced two at a time, in a random order. For each pair of edges, Painter immediately and irrevocably chooses one of the two possibilities to color one of them red and the other one blue. His goal is to avoid creating a monochromatic copy of a small fixed graph F for as long as possible. We show that the duration of the game is determined by a threshold function m H = m H ( n ) for certain graph-theoretic structures, e.g., cycles. That is, for every graph H in this family, Painter will asymptotically almost surely (a.a.s.) lose the game after m = ω ( m H ) edge pairs in the process. On the other hand, there exists an essentially optimal strategy: if the game lasts for m = o ( m H ) moves, Painter can a.a.s. successfully avoid monochromatic copies of H . Our attempt is to determine the threshold function for several classes of graphs.

Small subgraphs of random regular graphs

The main aim of this short paper is to answer the following question. Given a fixed graph H, for which values of the degree d does a random d-regular graph on n vertices contain a copy of H with probability close to one?

Rainbow $H$-factors

An $H$-factor of a graph $G$ is a spanning subgraph of $G$ whose connected components are isomorphic to $H$. Given a properly edge-colored graph $G$, a rainbow $H$-subgraph of $G$ is an $H$-subgraph of $G$ whose edges have distinct colors. A rainbow $H$-factor is an $H$-factor whose components are rainbow $H$-subgraphs. The following result is proved. If $H$ is any fixed graph with $h$ vertices then every properly edge-colored graph with $hn$ vertices and minimum degree $(1-1/\chi(H))hn+o(n)$ has a rainbow $H$-factor.

No Quantum Ergodicity for Star Graphs

We investigate statistical properties of the eigenfunctions of the Schrodinger operator on families of star graphs with incommensurate bond lengths. We show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average. We further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds. We describe how to construct such subsequences explicitly. These constructions are analogous to scars on short unstable periodic orbits.

Admission Control to Minimize Rejections

Admission control (call control) is a well-studied online problem. We are given a fixed graph with edge capacities, and must process a sequence of calls that arrive over time, accepting some and rejecting others in order to stay within capacity limitations of the network. In the standard theoretical formulation, this problem is analyzed as a benefit problem: The goal is to devise an online algorithm that accepts at least a reasonable fraction of the maximum number of calls that could possibly have been accepted in hindsight. This formulation, however, has the property that even algorithms with optimal competitive ratios (typically O(log n) where n is the number of nodes) may end up rejecting the vast majority of calls even when it would have been possible in hindsight to reject only very few. In this paper, we instead consider the goal of approximately minimizing the number of calls rejected. This is much more appropriate for settings in which rejections are intended to be rare events. In order to avoid t...

Coloring the hypergraph of maximal cliques of a graph with no long path

We consider the problem of coloring the vertices of a graph so that no maximal clique of size at least two is monocolored. We solve the following question: Given a fixed graph F, does there exist an integer f( F) such that the hypergraph of maximal cliques of any F-free graph can be f( F)-colored? We show that the answer is positive if and only if all components of F are paths. In that case we give an estimate of f( F), and the proof contains a polynomial time algorithm to find such a coloring.

Integer and Fractional Packings in Dense Graphs

Let be any fixed graph. For a graph G we define to be the maximum size of a set of pairwise edge-disjoint copies of in G. We say a function from the set of copies of in G to [0, 1] is a fractional -packing of G if for every edge e of G. Then is defined to be the maximum value of over all fractional -packings of G. We show that for all graphs G.

Graph Decomposition is NP-Complete: A Complete Proof of Holyer's Conjecture

An H-decomposition of a graph G=(V,E) is a partition of E into subgraphs isomorphic to H. Given a fixed graph H, the H-decomposition problem is to determine whether an input graph G admits an H-decomposition. In 1980, Holyer conjectured that H-decomposition is NP-complete whenever H is connected and has three edges or more. Some partial results have been obtained since then. A complete proof of Holyer's conjecture is the content of this paper. The characterization problem of all graphs H for which H-decomposition is NP-complete is hence reduced to graphs where every connected component contains at most two edges.