Further results on graph equations for line graphs and n-th power graphs

We present solutions of seven graph equations involving the line graph, complement and n-th power operations. One such equation $\text{L(G)}{}^{\mathrm{n}}\text{=}\text{G}\text{?}$ generalizes a result of M. Aigner. In addition, some comments are made about graphs satisfying $\text{G}{}^{\mathrm{n}}\text{=}\text{G}\text{?}$.     

A theoretical analysis of backtracking in the graph coloring problem

The graph coloring problem is: Given a positive integer K and a graph G. Can the vertices of G be properly colored in K colors? The problem is NP-complete. The average behavior of the simplest backtrack algorithm for this problem is studied. Average run time over all graphs is known to be bounded. Average run time over all graphs with n vertices and q edges behaves like $\text{exp}\text{(}\text{Cn}{}^2\text{q}\text{)}$. It is shown that similar results hold for all higher moments of the run time distribution. For all graphs and for graphs where lim $\text{n}{}^2\text{q}$ exists, the run time has a limiting disibution as n → ∞.     

Graphes orientes indécomposables en circuits hamiltoniens

Let G be a cubic graph of order 2n consisting of a cycle plus a perfect matching and let $\text{G}{}^1$ be the symmetric digraph obtained from G by replacing each edge of G by two opposite arcs. In this paper we study when $\text{G}{}^1$ can be decomposed into three hamiltonian circuits and in particular we prove that such a decomposition is impossible if n is even.     

Uniqueness and faithfulness of embedding of toroidal graphs

A topological generalization of the uniqueness of duals of 3-connected planar graphs will be obtained. A graph G is uniquely embeddable in a surface F if for any two embeddings $\text{?}{}_1$, $\text{?}{}_2\text{:G → F}$, there are an autohomeomorphism h:F → F and an automorphism σ:G → G such that $\text{h°?}{}_1\text{= ?}{}_2\text{°σ}$. A graph G is faithfully embedabble in a surface F if there is an embedding $\text{?:G → F}$ such that for any automorphism σ:G → G, there is an autohomeomorphism h:F → F with $\text{h°? = f°σ}$. Our main theorems state that any 6-connected toroidal graph is uniquelly embeddable in a torus and that any 6-connected toroidal graph with precisely three exceptions is faithfully embeddable in a torus. The proofs are based on a classification of 6-regular torus graphs.     

Random near-regular graphs and the node packing problem

Nemhauser and Trotter [12] proposed a certain easily-solved linear program as a relaxation of the node packing problem. They showed that any variables receiving integer values in an optimal solution to this linear program also take on the same values in an optimal solution to the (integer) node packing problem. Let π be the property of graphs defined as follows: a graph G has property π if and only if there is a unique optimal solution to the linear-relaxation problem, and this solution is completely fractional. If a graph G has property π then no information about the node packing problem on G is gained by solving the linear relaxation. We calculate the asymptotic probability that a certain type of ‘sparse’ random graph has property π, as the number of its nodes tends to infinity. Let m be a fixed positive integer, and consider the following random graph on the node set {1,2 …, n}). We join each node, j say, to exactly m other nodes chosen randomly with replacement, by edges oriented away from j; we denote by G_n(m) the undirected graph obtained by deleting all orientations and allowing all parallel edges to coalesce. We show that, as n → ∞,

$\text{P(G}{}_{\mathrm{n}}\text{(m)}\text{has property}\text{π)→}\text{0}\text{if}\text{m = 1,}\text{1}\text{if}\text{m ? 3,}$

and we conjecture that $\text{P(G}{}_{\mathrm{n}}\text{(2)}\text{has property}\text{π)→ (1–2e}{}^{\mathrm{?2}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.     

Derivations on a C1-algebra and its double dual

Let A be a C¹-algebra and X a Banach A-module. The module action of A on X gives rise to module actions of $\text{A}{}^7$ on $\text{X}{}^1$ and $\text{X}{}^7$, and derivations of A into X (resp. $\text{X}{}^1$) extend to derivations of $\text{A}{}^7$ into $\text{X}{}^7$ (resp. $\text{X}{}^1$). If A is nuclear, and X is a dual Banach A-module with $\text{X}{}^1$ weakly sequentially complete, then every derivation of A into X is inner. Under the same hypothesis on A, the extension to the finite part of $\text{A}{}^7$ of any derivation of A into any dual Banach A-module is inner, as are all derivations of A into $\text{A}{}^1$. Every derivation of a semifinite von Neumann algebra into its predual is inner.     

Sufficiency and invariance

Suppose that a statistical decision problem is invariant under a group of transformations g?G. T (X) is equivariant if there exists $\text{g}{}^1\text{? G}{}^1$ such that $\text{T(g(X)) = g}{}^1\text{(T((X))}$. We show that the minimal sufficient statistic is equivalent and that if T(X) is an equivariant sufficient statistics and d(X) is invariant under G, then $\text{d}{}^1\text{(T) = Ed(X)∥T}$ is invariant under $\text{G}{}^1$.     

Fast algorithms for generating all maximal independent sets of interval,circular-arc and chordal graphs

In this paper we give fast algorithms for generating all maximal independent sets of three special classes of graphs—interval, circular-arc, and chordal graphs. The worst-case running times of our algorithms are O(n² + β) for interval and circular-arc graphs, and $\text{O((n + e)}{}^1\text{α)}$ for chordal graphs, where n, e, and α are the numbers of vertexes, edges, and maximal independent sets of a graph, and β is the sum of the numbers of vertexes of all maximal independent sets. Our algorithms compare favorably with the fastest known algorithm for general graphs which has a worst-case running time of $\text{O(n}{}^1\text{e}{}^1\text{α)}$.     

Lower bounds for combinatorial problems on graphs

Nontrivial lower bounds are given for the computation time of various combinatorial problems on graphs under a linear or algebraic decision tree model. An $\text{Ω(n}{}^3\text{log}\text{n)}$ bound is obtained for a “travelling salesman problem” on a weighted complete graph of n vertices. Moreover it is shown that the same bound is valid for the “subgraph detection problem” with respect to property P if P is hereditary and determined by components. Thus an $\text{Ω(n}{}^3\text{log}\text{n)}$ bound is established in a unified way for a rather large class of problems.     

Factorizing the complete graph into factors with large star number

The graph G has star number n if any n vertices of G belong to a subgraph which is a star. Let f(n, k) be the smallest number m such that the complete graph on m vertices can be factorized into k factors with star number n. In the present paper we prove that $\text{c}{}_1{}^{\mathrm{n}}\text{k ≤ f(n, k) < c}{}_2{}^{\mathrm{n}}\text{k}$.     

Solution of the Hamiltonian problem for self-complementary graphs

A graph G is said to be highly constricted if there exists a nonempty subset S of vertices such that (i) G ? S has more than |S| components, (ii) S induces the complete graph, and (iii) for every u ∈ S and v ? S, we have d_G(u) > d_G(v), where d_G(u) denotes the degree of u in G. In this paper it is shown that a non-hamiltonian self-complementary graph G of order p is highly constricted, unless p = 4N and G is a particular graph $\text{G}{}^1\text{(4N)}$. It is also proved that if G is a self-complementary graph of order p(≥8) and π its degree sequence, then G is pancyclic if π has a realization with a hamiltonian cycle, and G has a 2-factor if π has a realization with a 2-factor, unless p = 4N and $\text{G = G}{}^1\text{(4N)}$.     

Minimally k-connected graphs of low order and maximal size

Let G be a minimally k-connected graph of order n and size e(G).Mader [4] proved that (i) $\text{e(G)?kn?(}\text{k+1}\text{2}\text{)}$; (ii) e(G)?k(n?k) if n?3k?2, and the complete bipartite graph K_k,n?k is the only minimally k-connected graph of order; n and size k(n?k) when k?2 and n?3k?1.The purpose of the present paper is to determine all minimally k-connected graphs of low order and maximal size. For each n such that k+1?n?3k?2 we prove $\text{e(G)??}\text{(n+k)}{}^2\text{8}\text{?}$ and characterize all minimally k-connected graphs of order n and size $\text{?(}\text{(n+k)}{}^2\text{8}\text{?}$.     

Groups with completely regular primitive dual space

We show that if G is a unimodular, amenable group whose primitive dual space Pr $\text{C}{}^1\text{(G)}$ satisfies certain regularity conditions, then G is in the class [FC]^?. This is a partial converse to a result of Liukkonen and Mosak that a σ-compact [FC]^? group has a Hausdorff primitive dual space.     

Balancing signed graphs

A signed graph based on F is an ordinary graph F with each edge marked as positive or negative. Such a graph is called balanced if each of its cycles includes an even number of negative edges. Psychologists are sometimes interested in the smallest number d=d(G) such that a signed graph G may be converted into a balanced graph by changing the signs of d edges. We investigate the number D(F) defined as the largest d(G) such that G is a signed graph based on F. We prove that $\text{1}\text{2}\text{m?}\text{nm}\text{≤D(F)≤}\text{1}\text{2}\text{m}$ for every graph F with n vertices and m edges. If F is the complete bipartite graph with t vertices in each part, then $\text{D(F)≤}\text{1}\text{2}\text{t}{}^2\text{?ct}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ for some positive constant c.     

Some pathologies in the Mackey analysis for a certain nonseparable group

Suppose G is a separable locally compact group and N is a closed normal subgroup. If the dual N? is smooth and the orbit space $\text{N}\text{?}\text{G}$ is smooth for the natural action of G on $\text{N}\text{?}\text{(L}{}^{\mathrm{x}}\text{(n) = L(xnx}{}^{\mathrm{?1}}\text{))}$, the method of G. W. Mackey (Acta Math.99 (1958), 265–311) gives a fairly simple procedure for constructing the dual ?. In this paper we examine an example which shows that the nonseparable case is much more complicated. In the example, N is abelian, $\text{N}\text{?}\text{G}$ is finite and even when the stabilizer is N there are many irreducible representations of G associated with the same orbit.     

Positive definite distributions on K\GK

Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space $\text{C}$¹(G). We show that the corresponding property is no longer true for the space of double cosets $\text{K}\text{G}\text{K}$. If G is of real-rank 1, we construct liner functionals $\text{T}{}_{\mathrm{p}}\text{? (C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{))′}$ for each p, 0 < p ? 2, such that $\text{T}{}_{\mathrm{p}}\text{(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$ but T_p does not extend to a continuous functional on $\text{C}{}^{\mathrm{p}}\text{(K}\text{G}\text{K}\text{)}$. In particular, if p ? 1, T_v does not extend to a continuous functional on $\text{C}{}^1\text{(K}\text{G}\text{K}\text{)}$. We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that $\text{T(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$. The main tool used is a theorem of Trombi-Varadarajan.     

Comparability graphs and intersection graphs

A function diagram (f-diagram) D consists of the family of curves {$\text{1}\text{?}$ñ} obtained from n continuous functions $\text{f}{}_{\mathrm{i}}\text{:[0,1]→}\text{R}\text{(1?i?n)}$. We call the intersection graph of D a function graph (f-graph). It is shown that a graph G is an f-graph if and only if its complement ? is a comparability graph. An f-diagram generalizes the notion of a permulation diagram where the f_i are linear functions. It is also shown that G is the intersection graph of the concatenation of ?k permutation diagrams if and only if the partial order dimension of $\text{G}\text{?}\text{is ?k+1}$. Computational complexity results are obtained for recognizing such graphs.     

Planar cubic hypohamiltonian and hypotraceable graphs

Infinite families of planar cubic hypohamiltonian and hypotraceable graphs are described and these are used to prove that the maximum degree and the maximum number of edges in a hypohamiltonian graph with n vertices are approximately $\text{n}\text{2}$ and $\text{n}{}^2\text{4}$, respectively. Also, the possible order of a cubic hypohamiltonian graph is determined.     

Solving NP-hard problems in ‘almost trees’: Vertex cover

We present an algorithm which finds a minimum vertex cover in a graph G(V, E) in time $\text{O}\text{(|V|+(}\text{a}\text{k}\text{)2}{}^{\mathrm{<mtext>k</mtext><mtext>3</mtext>}}\text{)}$, where for connected graphs G the parameter a is defined as the minimum number of edges that must be added to a tree to produce G, and k is the maximum a over all biconnected components of the graph. The algorithm combines two main approaches for coping with NP-completeness, and thereby achieves better running time than algorithms using only one of these approaches.