Addendum to ‘avoiding a giant component’

We take this opportunity, which is kindly provided by the editors, to add a theorem and to correct the proof of one of the lemmas in the article ‘Avoiding a Giant Component’ [Bohman and Frieze, Avoiding a giant component, Random Struct Alg 19 (2001), 75–85]. The subject of the said article is the following guided random process. Let e₁, $e^{\prime}_{1}$; e₂, $e^{\prime}_{2}$;…;e_i, $e^{\prime}_{i}$;… be a sequence of ordered pairs of edges chosen uniformly at random from the edge set of the complete graph K_n. This sequence is used to form a graph by choosing at stage i, i=1,…, one edge from e_i, $e_{i}^{\prime}$ to be an edge in the graph, where the choice at stage i is based only on the observation of the edges that have appeared by stage i. In Bohman and Frieze [Avoiding a giant component, Random Struct Alg 19 (2001), 75–85], we proved that these choices can be made so that whp (A sequence of events ?_n is said to occur with high probability (whp) if lim_n→∞ Pr (?_n)=1.), the size of the largest component of the graph formed at stage. 535n, is polylogarithmic in n. Here, we make a correction to that proof and prove that the graph formed at stage cn for any constant c>1 will contain a component of size Ω(n) (regardless of how the edges are chosen at each stage). © 2002 John Wiley & Sons, Inc. Random Struct. Alg., 20, 126–130, 2002     

A phase transition for avoiding a giant component

Let c be a constant and (e₁,f₁),(e₂,f₂),…,(e_cn,f_cn) be a sequence of ordered pairs of edges from the complete graph K_n chosen uniformly and independently at random. We prove that there exists a constant c₂ such that if c > c₂, then whp every graph which contains at least one edge from each ordered pair (e_i,f_i) has a component of size Ω(n), and, if c < c₂, then whp there is a graph containing at least one edge from each pair that has no component with more than O(n^1?? vertices, where ? is a constant that depends on c₂ ? c. The constant c₂ is roughly 0.97677. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

On the number of complementary trees in a graph

Consider a graph with no loops or multiple arcs with n+1 nodes and 2n arcs labeled a_l,…,a_n,a′_l,…,a′_n, where n ≥ 5. A spanning tree of such a graph is called complementary if it contains exactly one arc of each pair {a_i,a′_i}. The purpose of this paper is to develop a procedure for finding complementary trees in a graph, given one such tree. Using the procedure repeatedly we give a constructive proof that every graph of the above form which has one complementary tree has at least six such trees.     

Facial non‐repetitive edge‐coloring of plane graphs

A sequence r₁, r₂, …, r_2n such that r_i=r_{n+ i} for all 1≤i≤n is called a repetition. A sequence S is called non‐repetitive if no block (i.e. subsequence of consecutive terms of S) is a repetition. Let G be a graph whose edges are colored. A trail is called non‐repetitive if the sequence of colors of its edges is non‐repetitive. If G is a plane graph, a facial non‐repetitive edge‐coloring of G is an edge‐coloring such that any facial trail (i.e. a trail of consecutive edges on the boundary walk of a face) is non‐repetitive. We denote π′_f(G) the minimum number of colors of a facial non‐repetitive edge‐coloring of G. In this article, we show that π′_f(G)≤8 for any plane graph G. We also get better upper bounds for π′_f(G) in the cases when G is a tree, a plane triangulation, a simple 3‐connected plane graph, a hamiltonian plane graph, an outerplanar graph or a Halin graph. The bound 4 for trees is tight. © 2010 Wiley Periodicals, Inc. J Graph Theory 66: 38–48, 2010     

Multiplicative infinite loop space theory

Let G be a finitely presented group given by its pre-abelian presentation <X1,…,Xm; Xe11ζ1,…,Xemmζ,ζm+1,…>, where ei≥0 for i = 1,…, m and ζj?G′ for j≥1. Let N be the subgroup of G generated by the normal subgroups [xeii, G] for i = 1,…, m. Then Dn+2(G)≡γn+2(G) (modNG′) for all n≥0, where G” is the second commutator subgroup of G,γn+2(G) is the (n+2)th term of the lower central series of G and Dn+2(G) = G∩(1+△n+2(G)) is the (n+2)th dimension subgroup of G.     

The degree sequence of a random graph. I. The models

We show that the joint distribution of the degrees of a random graph can be accurately approximated by several simpler models derived from a set of independent binomial distributions. On the one hand, we consider the distribution of degree sequences of random graphs with n vertices and ½m edges. For a wide range of values of m, this distribution is almost everywhere in close correspondence with the conditional distribution {(X₁,…,X_n) | ∑ X_i=m}, where X₁,…,X_n are independent random variables, each having the same binomial distribution as the degree of one vertex. We also consider random graphs with n vertices and edge probability p. For a wide range of functions p=p(n), the distribution of the degree sequence can be approximated by {(X₁,…,X>_n) | ∑ X_i is even}, where X₁,…,X_n are independent random variables each having the distribution Binom (n−1, p′), where p′ is itself a random variable with a particular truncated normal distribution. To facilitate computations, we demonstrate techniques by which statistics in this model can be inferred from those in a simple model of independent binomial random variables. Where they apply, the accuracy of our method is sufficient to determine asymptotically all probabilities greater than n^−k for any fixed k. In this first paper, we use the geometric mean of degrees as a tutorial example. In the second paper, we will determine the asymptotic distribution of the tth largest degree for all functions t=t(n) as n→∞. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 97–117 (1997)     

Mutually orthogonal graph squares

A decomposition ??={G₁, G₂,…,G_s} of a graph G is a partition of the edge set of G into edge‐disjoint subgraphs G₁, G₂,…,G_s. If G_i?H for all i∈{1, 2, …, s}, then ?? is a decomposition of G by H. Two decompositions ??={G₁, G₂, …, G_n} and ?={F₁, F₂,…,F_n} of the complete bipartite graph K_n,n are orthogonal if |E(G_i)∩E(F_j)|=1 for all i,j∈{1, 2, …, n}. A set of decompositions {??₁, ??₂, …, ??_k} of K_{n, n} is a set of k mutually orthogonal graph squares (MOGS) if ??_i and ??_j are orthogonal for all i, j∈{1, 2, …, k} and i≠j. For any bipartite graph G with n edges, N(n, G) denotes the maximum number k in a largest possible set {??₁, ??₂, …, ??_k} of MOGS of K_{n, n} by G. El‐Shanawany conjectured that if p is a prime number, then N(p, P_{p+ 1})=p, where P_{p+ 1} is the path on p+ 1 vertices. In this article, we prove this conjecture. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 369–373, 2009     

Triangular embeddings of K((i-2) n,n,…,n)

For complete i-partite graphs of the form K(n₁, n, n, …, n) the largest value of n₁ that allows the graph to be triangularly-embedded into a surface is (i-2)n. In this paper the author constructs triangular embeddings into surfaces of some complete partite graphs of the form K((i-2)n, n, …, n). The embeddings are exhibited using embedding schemes but the surfaces into which K((i-2)n, n, …, n) are triangularly embedded can be seen to be particularly nice branched covers of a surface into which K(i-2, 1, 1,…,1) is triangularly embedded.     

An improved linear edge bound for graph linkages

A graph is said to be k-linked if it has at least 2k vertices and for every sequence s₁,…,s_k,t₁,…,t_k of distinct vertices there exist disjoint paths P₁,…,P_k such that the ends of P_i are s_i and t_i. Bollobás and Thomason showed that if a simple graph G on n vertices is 2k-connected and G has at least 11kn edges, then G is k-linked. We give a relatively simple inductive proof of the stronger statement that 8kn edges and 2k-connectivity suffice, and then with more effort improve the edge bound to 5kn.     

Prime links in some skew-polynomial and skew-laurent rings

Let R be a Noetherian commutative ring and a α₁,…,α_n commuting automorphisms of R. Define T = R[θ₁,…,θ_n;α₁,…,α_n] to be the skew-polynomial ring with θ_ir = α_i(r)θ_i and θ_iθ_j= θ_jθ_i, for all i,j ? (1,…,n) and r ? R, and let S = Rθ₁,θ₁:^-1,…,θ_n:,θ_n;^-1;α₁:,…,α_n] be the corresponding skew-Laurent ring. In this paper we show that S and T satisfy the strong second layer condition and characterize the links between prime ideals in these rings.     

Embeddings of bipartite graphs

If G is a bipartite graph with bipartition A, B then let G_m,n(A, B) be obtained from G by replacing each vertex a of A by an independent set a₁, …, a_m, each vertex b of B by an independent set b₁,…, b_n, and each edge ab of G by the complete bipartite graph with edges a_ib_j (1 ≤ i ≤ m and 1 ≤ j ≤ n). Whenever G has certain types of spanning forests, then cellular embeddings of G in surfaces S may be lifted to embeddings of G_m,n(A, B) having faces of the same sizes as those of G in S. These results are proved by the technique of “excess-current graphs.” They include new genus embeddings for a large class of bipartite graphs.     

Orientations of circle graphs

Let C(v₁, …,v_n) be a system consisting of a circle C with chords v₁, …,v_n on it having different endpoints. Define a graph G having vertex set V(G) = {v₁, …,v_n} and for which vertices v_i and v_j are adjacent in G if the chords v_i and v_j intersect. Such a graph will be called a circle graph. The chords divide the interior of C into a number of regions. We give a method which associates to each such region an orientation of the edges of G. For a given C(v₁, …,v_n) the number m of different orientations corresponding to it satisfies q + 1 ≤ m ≤ n + q + 1, where q is the number of edges in G. An oriented graph obtained from a diagram C(v₁, …,v_n) as above is called an oriented circle graph (OCG). We show that transitive orientations of permutation graphs are OCGs, and give a characterization of tournaments which are OCGs. When the region is a peripheral one, the orientation of G is acyclic. In this case we define a special orientation of the complement of G, and use this to develop an improved algorithm for finding a maximum independent set in G.     

Doubly eulerian trails on rectangular grids

Let K be the 1-skeleton of the regular tessellation of Euclidean n-space by n-cubes, n ≥ 2. We show that K admits a doubly Eulerian trail (simply Eulerian trail), that is, a doubly infinite path π = … e_?1e₀e₁ … where, out of each pair {e, e^?1} of oppositely directed edges, both (exactly one) appear(s) exactly once in π, and where no e_i+1 = e_i^?1 (there are no U-turns).     

Additive and hereditary properties of graphs are uniquely factorizable into irreducible factors

A hereditary property of graphs is any class of graphs closed under isomorphism and subgraphs. Let 𝒫₁, 𝒫₂,…, 𝒫_n be hereditary properties of graphs. We say that a graph G has property 𝒫_1°𝒫_2°···_°𝒫_n if the vertex set of G can be partitioned into n sets V₁, V₂,…, V_n such that the subgraph of G induced by V_i belongs to 𝒫_i; i = 1, 2,…, n. A hereditary property is said to be reducible if there exist hereditary properties 𝒫₁ and 𝒫₂ such that ℛ = 𝒫_1°𝒫₂; otherwise it is irreducible. We prove that the factorization of a reducible hereditary property into irreducible factors is unique whenever the property is additive, i.e., it is closed under the disjoint union of graphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 44–53, 2000     

Gotzmann Edge Ideals

Let P = 𝕜[x ₁,…, x _n ] be the polynomial ring in n variables. A homogeneous ideal I ? P generated in degree d is called Gotzmann if it has the smallest possible Hilbert function out of all homogeneous ideals with the same dimension in degree d. The edge ideal of a simple graph G on vertices x ₁,…, x _n is the quadratic square-free monomial ideal generated by all x _i x _j where {x _i , x _j } is an edge of G. The only edge ideals that are Gotzmann are those edge ideals corresponding to star graphs.     

Vertex-disjoint quadrilaterals containing specified edges in a bipartite graph

The theory of vertex-disjoint cycles and 2-factors of graphs is the extension and generation of the well-known Hamiltonian cycles theory and it has important applications in network communication. In this paper we first prove the following result. Let G=(V ₁,V ₂;E) be a bipartite graph with |V ₁|=|V ₂|=n such that n≥2k+1, where k≥1 is an integer. If d(x)+d(y)≥?(4n+2k?1)/3? for each pair of nonadjacent vertices x and y of G with x∈V ₁ and y∈V ₂, then, for any k independent edges e ₁,…,e _k of G, G contains k vertex-disjoint quadrilaterals C ₁,…,C _k such that e _i ∈E(C _i ) for each i∈{1,…,k}. We further show that the degree condition above is sharp. If |V ₁|=|V ₂|=2k, we give degree conditions that G has a 2-factor with k vertex-disjoint quadrilaterals C ₁,…,C _k containing specified edges of G.     

Conservative graphs

A graph G with q edges is defined to be conservative if the edges of G can be oriented and distinctly numbered with the integers 1, 2,…, q so that at each vertex the sum of the numbers on the inwardly directed edges equals that on the outwardly directed edges. Several classes of graphs, including K_n, for n ≥4, and K_{2n, 2m}, for n, m ≥ 2, are shown to be conservative. It is proven that the dual of a planar graceful graph is conservative, and that the converse of this result is false.