Orientability thresholds for random hypergraphs

Let $h>w>0$ be two fixed integers. Let \orH be a random hypergraph whose hyperedges are all of cardinality $h$. To {\em $w$-orient} a hyperedge, we assign exactly $w$ of its vertices positive signs with respect to the hyperedge, and the rest negative. A $(w,k)$-orientation of \orH consists of a $w$-orientation of all hyperedges of \orH, such that each vertex receives at most $k$ positive signs from its incident hyperedges. When $k$ is large enough, we determine the threshold of the existence of a $(w,k)$-orientation of a random hypergraph. The $(w,k)$-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The graph case, when $h=2$ and $w=1$, was solved recently by Cain, Sanders and Wormald and independently by Fernholz and Ramachandran, which settled a conjecture of Karp and Saks.Comment: 47 pages, 1 figures, the journal version of [16

Enumeration of graphs with a heavy-tailed degree sequence

In this paper, we asymptotically enumerate graphs with a given degree sequence d=(d_1,...,d_n) satisfying restrictions designed to permit heavy-tailed sequences in the sparse case (i.e. where the average degree is rather small). Our general result requires upper bounds on functions of M_k= \sum_{i=1}^n [d_i]_k for a few small integers k\ge 1. Note that M_1 is simply the total degree of the graphs. As special cases, we asymptotically enumerate graphs with (i) degree sequences satisfying M_2=o(M_1^{ 9/8}); (ii) degree sequences following a power law with parameter gamma>5/2; (iii) power-law degree sequences that mimic independent power-law "degrees" with parameter gamma>1+\sqrt{3}\approx 2.732; (iv) degree sequences following a certain "long-tailed" power law; (v) certain bi-valued sequences. A previous result on sparse graphs by McKay and the second author applies to a wide range of degree sequences but requires Delta =o(M_1^{1/3}), where Delta is the maximum degree. Our new result applies in some cases when Delta is only barely o(M_1^ {3/5}). Case (i) above generalises a result of Janson which requires M_2=O(M_1) (and hence M_1=O(n) and Delta=O(n^{1/2})). Cases (ii) and (iii) provide the first asymptotic enumeration results applicable to degree sequences of real-world networks following a power law, for which it has been empirically observed that 2<gamma<3.Comment: 34 page

Rainbow Hamilton cycles in random regular graphs

A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n tends to infinity, provided d is at least 8.Comment: 16 page

Local algorithms, regular graphs of large girth, and random regular graphs

We introduce a general class of algorithms and supply a number of general results useful for analysing these algorithms when applied to regular graphs of large girth. As a result, we can transfer a number of results proved for random regular graphs into (deterministic) results about all regular graphs with sufficiently large girth. This is an uncommon direction of transfer of results, which is usually from the deterministic setting to the random one. In particular, this approach enables, for the first time, the achievement of results equivalent to those obtained on random regular graphs by a powerful class of algorithms which contain prioritised actions. As examples, we obtain new upper or lower bounds on the size of maximum independent sets, minimum dominating sets, maximum and minimum bisection, maximum $k$-independent sets, minimum $k$-dominating sets and minimum connected and weakly-connected dominating sets in $r$-regular graphs with large girth.Comment: Third version: no changes were made to the file. We would like to point out that this paper was split into two parts in the publication process. General theorems are in a paper with the same title, accepted by Combinatorica. The applications of Section 9 are in a paper entitled "Properties of regular graphs with large girth via local algorithms", published by JCTB, doi 10.1016/j.jctb.2016.07.00

On the Stretch Factor of Randomly Embedded Random Graphs

We consider a random graph G(n,p) whose vertex set V has been randomly embedded in the unit square and whose edges are given weight equal to the geometric distance between their end vertices. Then each pair {u,v} of vertices have a distance in the weighted graph, and a Euclidean distance. The stretch factor of the embedded graph is defined as the maximum ratio of these two distances, over all u,v in V. We give upper and lower bounds on the stretch factor (holding asymptotically almost surely), and show that for p not too close to 0 or 1, these bounds are best possible in a certain sense. Our results imply that the stretch factor is bounded with probability tending to 1 if and only if n(1-p) tends to 0, answering a question of O'Rourke.Comment: 12 page

Maximum edge-cuts in cubic graphs with large girth and in random cubic graphs

We show that for every cubic graph G with sufficiently large girth there exists a probability distribution on edge-cuts of G such that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that G contains an edge-cut of size at least 1.33008n, where n is the number of vertices of G, and has fractional cut covering number at most 1.127752. The lower bound on the size of maximum edge-cut also applies to random cubic graphs. Specifically, a random n-vertex cubic graph a.a.s. contains an edge cut of size 1.33008n