Rainbow perfect matchings and Hamilton cycles in the random geometric graph

Given a graph on n vertices and an assignment of colours to the edges, a rainbow Hamilton cycle is a cycle of length n visiting each vertex once and with pairwise different colours on the edges. Similarly (for even n) a rainbow perfect matching is a collection of independent edges with pairwise different colours. In this note we show that if we randomly colour the edges of a random geometric graph with sufficiently many colours, then a.a.s. the graph contains a rainbow perfect matching (rainbow Hamilton cycle) if and only if the minimum degree is at least 1 (respectively, at least 2). More precisely, consider n points (i.e. vertices) chosen independently and uniformly at random from the unit d‐dimensional cube for any fixed . Form a sequence of graphs on these n vertices by adding edges one by one between each possible pair of vertices. Edges are added in increasing order of lengths (measured with respect to the norm, for any fixed ). Each time a new edge is added, it receives a random colour chosen uniformly at random and with repetition from a set of colours, where a sufficiently large fixed constant. Then, a.a.s. the first graph in the sequence with minimum degree at least 1 must contain a rainbow perfect matching (for even n), and the first graph with minimum degree at least 2 must contain a rainbow Hamilton cycle. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 587–606, 2017     

Algorithms for graphs of bounded treewidth via orthogonal range searching

We show that, for any fixed constant k3, the sum of the distances between all pairs of vertices of an abstract graph with n vertices and treewidth at most k can be computed in O(nlog^k−1n) time.We also show that, for any fixed constant k2, the dilation of a geometric graph (i.e., a graph drawn in the plane with straight-line segments) with n vertices and treewidth at most k can be computed in O(nlog^k+1n) expected time. The dilation (or stretch-factor) of a geometric graph is defined as the largest ratio, taken over all pairs of vertices, between the distance measured along the graph and the Euclidean distance.The algorithms for both problems are based on the same principle: data structures for orthogonal range searching in bounded dimension provide a compact representation of distances in abstract graphs of bounded treewidth.     

Explicit sparse almost‐universal graphs for ${\bf {{\cal G}(n, {k \over n})}}$

For any integer n, let be a probability distribution on the family of graphs on n vertices (where every such graph has nonzero probability associated with it). A graph Γ is ‐almost‐universal if Γ satisifies the following: If G is chosen according to the probability distribution , then G is isomorphic to a subgraph of Γ with probability 1 ‐ . For any p ∈ [0,1], let (n,p) denote the probability distribution on the family of graphs on n vertices, where two vertices u and v form an edge with probability p, and the events {u and v form an edge}; u,v ∈ V (G) are mutually independent. For k ≥ 4 and n sufficiently large we construct a ‐almost‐universal‐graph on n vertices and with O(n)polylog(n) edges, where q = ? ? for such k ≤ 6, and where q = ? ? for k ≥ 7. The number of edges is close to the lower bound of Ω( ) for the number of edges in a universal graph for the family of graphs with n vertices and maximum degree k. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Paths with restricted degrees of their vertices in planar graphs

In this paper it is proved that every 3-connected planar graph contains a path on 3 vertices each of which is of degree at most 15 and a path on 4 vertices each of which has degree at most 23. Analogous results are stated for 3-connected planar graphs of minimum degree 4 and 5. Moreover, for every pair of integers n 3, k 4 there is a 2-connected planar graph such that every path on n vertices in it has a vertex of degree k.     

The isoperimetric constant of the random graph process

The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of , taken over all nonempty subsets S ⊂ V (G) of size at most n/2, where ∂S denotes the set of edges with precisely one end in S. A random graph process on n vertices, , is a sequence of graphs, where is the edgeless graph on n vertices, and is the result of adding an edge to , uniformly distributed over all the missing edges. The authors show that in almost every graph process equals the minimal degree of as long as the minimal degree is o(log n). Furthermore, it is shown that this result is essentially best possible, by demonstrating that along the period in which the minimum degree is typically Θ(log n), the ratio between the isoperimetric constant and the minimum degree falls from 1 to , its final value. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

On the total variation distance between the binomial random graph and the random intersection graph

When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karoński, Scheinerman, and Singer‐Cohen introduced a random intersection graph by taking randomly assigned sets. The random intersection graph has n vertices and sets assigned to the vertices are chosen to be i.i.d. random subsets of a fixed set M of size m where each element of M belongs to each random subset with probability p, independently of all other elements in M. In 2000, Fill, Scheinerman, and Singer‐Cohen showed that the total variation distance between the random graph and the Erdös‐Rényi graph tends to 0 for any if , where is chosen so that the expected numbers of edges in the two graphs are the same. In this paper, it is proved that the total variation distance still tends to 0 for any whenever .     

Sparse partition universal graphs for graphs of bounded degree

In 1983, Chvátal, Trotter and the two senior authors proved that for any Δ there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph KN with N?Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Δ. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and edges, with N=⌈B^′n⌉ for some constant B^′ that depends only on Δ. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Δ is . Our approach is based on random graphs; in fact, we show that the classical Erd?s–Rényi random graph with the numerical parameters above satisfies a stronger partition property with high probability, namely, that any 2-colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree Δ.The main tool in our proof is the regularity method, adapted to a suitable sparse setting. The novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs. Crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed.     

On the Circumference of 3-Connected Quasi-Claw-Free Graphs

A graph G is quasi-claw-free if it satisfies the property: d(x,y)=2 ⇒ there exists such that . In the paper, we prove that the circumference of a 3-connected quasi-claw-free graph G on n vertices is at least min{4δ−2,n} and G is hamiltonian if n≤5δ−5.     

On the number of hamilton cycles in a random graph

Let a random graph G be constructed by adding random edges one by one, starting with n isolated vertices. We show that with probability going to one as n goes to infinity, when G first has minimum degree two, it has at least (log n) distinct hamilton cycles for any fixed ?>0.     

Chordal 2‐Connected Graphs and Spanning Trees

We present a transformation on a chordal 2‐connected simple graph that decreases the number of spanning trees. Based on this transformation, we show that for positive integers n, m with , the threshold graph having n vertices and m edges that consists of an ‐clique and vertices of degree 2 is the only graph with the fewest spanning trees among all 2‐connected chordal graphs on n vertices and m edges.     

Bounds on the cover time

Consider a particle that moves on a connected, undirected graphG withn vertices. At each step the particle goes from the current vertex to one of its neighbors, chosen uniformly at random. Tocover time is the first time when the particle has visited all the vertices in the graph starting from a given vertex. In this paper, we present upper and lower bounds that relate the expected cover time for a graph to the eigenvalues of the Markov chain that describes the random walk above. An interesting consequence is that regular expander graphs have expected cover time (n logn).This research was done while this author was a postdoctoral fellow in the Department of Computer Science, Princeton University, and it was supported in part by DNR grant N00014-87-K-0467.     

A Characterization of Graphs with No Cube Minor

In this paper it is shown that any 4-connected graph that does not contain a minor isomorphic to the cube is a minor of the line graph of V_n for some n6 or a minor of one of five graphs. Moreover, there exists a unique 5-connected graph on at least 8 vertices with no cube minor and a unique 4-connected graph with a vertex of degree at least 8 with no cube minor. Further, it is shown that any graph with no cube minor is obtained from 4-connected such graphs by 0-, 1-, and 2-summing, and 3-summing over a specified triangles.     

A circular embedding of a graph in Euclidean 3-space

A spatial embedding of a graph G is an embedding of G into the 3-dimensional Euclidean space . J.H. Conway and C.McA. Gordon proved that every spatial embedding of the complete graph on 7 vertices contains a nontrivial knot. A linear spatial embedding of a graph is an embedding which maps each edge to a single straight line segment. In this paper, we construct a linear spatial embedding of the complete graph on 2n−1 (or 2n) vertices which contains the torus knot T(2n−5,2) (n4). A circular spatial embedding of a graph is an embedding which maps each edge to a round arc. We define the circular number of a knot as the minimal number of round arcs in among such embeddings of the knot. We show that a knot has circular number 3 if and only if the knot is a trefoil knot, and the figure-eight knot has circular number 4.     

Nearly uniform distribution of edges among k-subgraphs of a graph

We investigate the behavior of the function f = f(n, k, e) defined as the smallest integer with the following property: If in a graph on n vertices, the numbers of edges in any two induced subgraphs on k vertices differ by at most e, then the graph or its complement has at most f edges. One of the results states that . © 1929 John Wiley & Sons, Inc.     

Cycles through a prescribed vertex set in N-connected graphs

A well-known result of Dirac (Math. Nachr. 22 (1960) 61) says that given n vertices in an n-connected G, G has a cycle through all of them. In this paper, we generalize Dirac's result as follows:Given at most vertices in an n-connected graph G when n3 and , then G has a cycle through exactly n vertices of them.This improves the previous known bound given by Kaneko and Saito (J. Graph Theory 15(6) (1991) 655).     

A greedy algorithm for finding a large 2‐matching on a random cubic graph

A 2‐matching of a graph G is a spanning subgraph with maximum degree two. The size of a 2‐matching U is the number of edges in U and this is at least where n is the number of vertices of G and κ denotes the number of components. In this article, we analyze the performance of a greedy algorithm 2greedy for finding a large 2‐matching on a random 3‐regular graph. We prove that with high probability, the algorithm outputs a 2‐matching U with .     

Regular graphs whose subgraphs tend to be acyclic

Motivated by a problem that arises in the study of mirrored storage systems, we describe, for any fixed ε, δ > 0, and any integer d ≥ 2, explicit or randomized constructions of d‐regular graphs on n > n₀(ε, δ) vertices in which a random subgraph obtained by retaining each edge, randomly and independently, with probability , is acyclic with probability at least 1 ? δ. On the other hand we show that for any d‐regular graph G on n > n₁(ε, δ) vertices, a random subgraph of G obtained by retaining each edge, randomly and independently, with probability , does contain a cycle with probability at least 1 ? δ. The proofs combine probabilistic and combinatorial arguments, with number theoretic techniques. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Sparse universal graphs for bounded‐degree graphs

Let ℋ︁ be a family of graphs. A graph T is ℋ︁‐universal if it contains a copy of each H ∈ℋ︁ as a subgraph. Let ℋ︁(k,n) denote the family of graphs on n vertices with maximum degree at most k. For all positive integers k and n, we construct an ℋ︁(k,n)‐universal graph T with edges and exactly n vertices. The number of edges is almost as small as possible, as Ω(n^2‐2/k) is a lower bound for the number of edges in any such graph. The construction of T is explicit, whereas the proof of universality is probabilistic and is based on a novel graph decomposition result and on the properties of random walks on expanders. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Distance-2-Matchings of Random Graphs

In this paper we study the maximal size of a distance-2-matching in a random graph G _n;M , i.e., the probability space consisting of subgraphs of the complete graph over n vertices, K _n , having exactly M edges and uniform probability measure. A distance-2-matching in a graph Y, M ₂, is a set of Y-edges with the property that for any two elements every pair of their 4 incident vertices has Y-distance 2. Let M₂(Y) be the maximal size of a distance-2-matching in Y. Our main results are the derivation of a lower bound for M₂(Y) and a sharp concentration result for the random variable AMS Subject Classification: 05C80, 05C70.