Robert W. Robinson

Almost all regular graphs are hamiltonian

In a previous article the authors showed that almost all labelled cubic graphs are hamiltonian. In the present article, this result is used to show that almost all r‐regular graphs are hamiltonian for any fixed r ⩾ 3, by an analysis of the distribution of 1‐factors in random regular graphs. Moreover, almost all such graphs are r‐edge‐colorable if they have an even number of vertices. Similarly, almost all r‐regular bipartite graphs are hamiltonian and r‐edge‐colorable for fixed r ⩾ 3.

Almost all cubic graphs are Hamiltonian

In a previous article the authors showed that at least 98.4% of large labelled cubic graphs are hamiltonian. In the present article, this is improved to 100% in the limit by asymptotic analysis of the variance of the number of Hamilton cycles with respect to populations of cubic graphs with fixed numbers of short odd cycles.

Uniquely colorable graphs

Abstract A graph is called uniquely colorable if there is only one partition of its point set into the smallest possible number of color classes. Theorems concerning uniquely colorable graphs G include: o (1) The subgraph of G induced by the union of two color classes is connected. (2) Every homomorphic image of G is also uniquely colorable. (3) If a single new point w is added to G, and also at least one line joining w with each color class of G, then the resulting graph is uniquely colorable. (4) If u is a point of G of degree χ(G)−1, then G−u is uniquely colorable. A result concerning n-colorable graphs is strengthened by a proof that, for all n≥3, there exists a uniquely n-colorable graph not containing Kn.

The distribution of degrees in a large random tree

For labeled trees, Renyi showed that the probability that an arbitrary point of a random tree has degree k approaches l/e(k-l)!. For unlabeled trees, the answer is different because the number of ways to label a given tree depends on the order of its automorphism group. Using arguments involving combinatorial enumeration and asymptotics, we evaluate the corresponding probabilities for large unlabeled trees.

Enumeration of non-separable graphs*

Abstract Non-separable graphs are enumerated, and also graphs without end-points. The basic enumeration tool is sums of cycle indices of automorphism groups.

Isomorphic factorisations. I. Complete graphs

An isomorphic factorisation of the complete graph K. is a partition of the lines of Kp, into t isomorphic spanning subgraphs G; we then write GIK, and G E Kp/t. If the set of graphs KE/t is not empty, then of course tlp(p 1)/2. Our principal purpose is to prove the converse. It was found by Laura Guidotti that the converse does hold whenever (t, p) 1 or (t,p 1) = 1, We give a new and shorter proof of her result which involves permuting the points and lines of A.> The construction developed in our proof happens to give all the graphs in K6/3 and K7/3. The Divisibility Theorem asserts that there is a factorisation of K, into t isomorphic parts whenever t dividesp(p 1)/2. The proof to be given is based on our proof of Guidottis Theorem, with embellishments to handle the additional difficulties presented by the cases when t is not relatively prime top or p 1.

Twenty-step algorithm for determining the asymptotic number of trees of various speces

The technique for finding the asymptotic number of unlabeled trees of various sorts was developed by Polya (1937) and perfected by Otter (1948). Modern presentations are available in the book of Harary and Palmer (1973; Chapter 9), and in the paper of Bender (to appear). An exposition of the basic method is here developed in the form of a 20 step algorithm, which should facilitate the finding of asymptotic formulas for different kinds of trees. These 20 steps are presented in Section 2, and methods of justifying the steps are supplied in Section 3. In Sections 4, 5 and 6, the algorithm is applied to finding asymptotic values for the number of identity trees, homeomorphically irreducible trees, and a class of blocks with tree-like properties. The first two of these species were enumerated by Harary and Prins (1959) and the third is easily done. However, no asymptotic analyses have been given previously. For the purpose of the discussion in Sections 2 and 3, a hypothetical class lof trees is posed, of which there are Sn planted trees on n + 1 points (including [the root which is an endpoint; hence there are n lines) and sn unrooted trees on In points. We let S(x) and s(x) be the ordinary generating functions

Generating and Counting Hamilton Cycles in Random Regular Graphs

LetGbe chosen uniformly at random from the set G(r,n) ofr-regular graphs with vertex set n. We describe polynomial time algorithms thatwhp(i) find a Hamilton cycle inG, and (ii) approximately count the number of Hamilton cycles inG.

Asymptotic Enumeration of Eulerian Circuits in the Complete Graph

We determine the asymptotic behaviour of the number of Eulerian circuits in a complete graph of odd order. One corollary of our result is the following. If a maximum random walk, constrained to use each edge at most once, is taken on Kn, then the probability that all the edges are eventually used is asymptotic to e3/4n−½. Some similar results are obtained about Eulerian circuits and spanning trees in random regular tournaments. We also give exact values for up to 21 nodes.

The asymptotic number of acyclic digraphs

We obtain an asymptotic formula forAn,q, the number of digraphs withn labeled vertices,q edges and no cycles. The derivation consists of two separate parts. In the first we analyze the generating function forAn,q so as to obtain a central limit theorem for an associated probability distribution. In the second part we show combinatorially thatAn,q is a smooth function ofq. By combining these results, we obtain the desired asymptotic formula.