Partition identities and the coin exchange problem

The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a and b. This generalizes identities of MacMahon and Andrews. The analogous identities for three or more integers (in place of a,b) hold in certain cases.     

Asymmetric graph coloring games

We introduce the (a,b)‐coloring game, an asymmetric version of the coloring game played by two players Alice and Bob on a finite graph, which differs from the standard version in that, in each turn, Alice colors a vertices and Bob colors b vertices. We also introduce a related game, the (a,b)‐marking game. We analyze these games and determine the (a,b)‐chromatic numbers and (a,b)‐coloring numbers for the class of forests and all values of a and b. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 169–185, 2005     

Geometry of continued fractions associated with Frobenius numbers

The article describes the interrelations between the minimal integer number N(a,b,c) which belongs to the additive semigroup of integers generated by a, b, c together with all greater integers, on the one hand, and the geometrical theory of continued fractions describing the convex hulls of sets of integer points in simplicial cones, on the other hand. It also provides some hints on the extension of N to non-integral arguments.     

Packing two forests into a bipartite graph

For two integers a and b, we say that a bipartite graph G admits an (a, b)-bipartition if G has a bipartition (X, Y) such that |X| = a and |Y| = b. We say that two bipartite graphs G and H are compatible if, for some integers a and b, both G and H admit (a, b)-bipartitions. In this note, we prove that any two compatible trees of order n can be packed into a complete bipartite graph of order at most n + 1. We also provide a family of infinitely many pairs of compatible trees which cannot be packed into a complete bipartite graph of the same order. A theorem about packing two forests into a complete bipartite graph is derived from this result. © 1996 John Wiley & Sons, Inc.     

On the mean square of the error term for the asymmetric two-dimensional divisor problem (I)

Suppose a and b are two fixed positive integers such that (a, b) = 1. In this paper we shall establish an asymptotic formula for the mean square of the error term Δ _a,b (x) of the general two-dimensional divisor problem.     

Bounding the Derived Length for a Given Set of Character Degrees

Let G be a finite solvable group with {1, a, b, c, ab, ac} as the character degree set, where a ,b, and c are pairwise coprime integers greater than 1. We show that the derived length of G is at most 4. This verifies that the Taketa inequality, dl(G) ≤ |cd(G)|, is valid for solvable groups with {1, a, b, c, ab, ac} as the character degree set. Also, as a corollary, we conclude that if a, b, c, and d are pairwise coprime integers greater than 1 and G is a solvable group such that cd(G) = {1, a, b, c, d, ac, ad, bc, bd}, then dl(G) ≤ 5. Finally, we construct a family of solvable groups whose derived lengths are 4 and character degree sets are in the form {1, p, b, pb, q ^p , pq ^p }, where p is a prime, q is a prime power of an odd prime, and b > 1 is integer such that p, q, and b are pairwise coprime. Hence, the bound 4 is the best bound for the derived length of solvable groups whose character degree set is in the form {1, a, b, c, ab, ac} for some pairwise coprime integers a, b, and c.     

On the enumeration of parking functions by leading terms

Let x=(x₁,…,x_n) be a sequence of positive integers. An x-parking function is a sequence (a₁,…,a_n) of positive integers whose non-decreasing rearrangement b₁b_n satisfies b_ix₁++x_i. In this paper we give a combinatorial approach to the enumeration of (a,b,…,b)-parking functions by their leading terms, which covers the special cases x=(1,…,1), (a,1,…,1), and (b,…,b). The approach relies on bijections between the x-parking functions and labeled rooted forests. To serve this purpose, we present a simple method for establishing the required bijections. Some bijective results between certain sets of x-parking functions of distinct leading terms are also given.     

Asymmetric directed graph coloring games

This note generalizes the (a,b)-coloring game and the (a,b)-marking game which were introduced by Kierstead [H.A. Kierstead, Asymmetric graph coloring games, J. Graph Theory 48 (2005) 169-185] for undirected graphs to directed graphs. We prove that the (a,b)-chromatic and (a,b)-coloring number for the class of orientations of forests is b+2 if b≤a, and infinity otherwise. From these results we deduce upper bounds for the (a,b)-coloring number of oriented outerplanar graphs and of orientations of graphs embeddable in a surface with bounded girth.     

Extremal values of continuants and transcendence of certain continued fractions

We prove a criterion for the transcendence of continued fractions whose partial quotients are contained in a finite set {b₁,…,b_r} of positive integers such that the density of occurrences of b_i in the sequence of partial quotients exists for 1ir. As an application we study continued fractions [0,a₁,a₂,a₃,…] with a_n=1+([nθ]modd) where θ is irrational and d2 is a positive integer.     

Decomposition of Complete Bipartite Even Graphs into Closed Trails

We prove that any complete bipartite graph K _a,b , where a, b are even integers, can be decomposed into closed trails with prescribed even lengths.     

[a,b]-factorization of a graph

Let a and b be integers such that 0 ? a ? b. Then a graph G is called an [a, b]-graph if a ? d_G(x) ? b for every x ? V(G), and an [a, b]-factor of a graph is defined to be its spanning subgraph F such that a ? d_F(x) ? b for every vertex x, where d_G(x) and d_F(x) denote the degrees of x in G and F, respectively. If the edges of a graph can be decomposed into [a.b]-factors then we say that the graph is [2a, 2a]-factorable. We prove the following two theorems: (i) a graph G is [2a, 2b)-factorable if and only if G is a [2am,2bm]-graph for some integer m, and (ii) every [8m + 2k, 10m + 2k]-graph is [1,2]-factorable.     

Disjunctive Rado numbers

If L₁ and L₂ are linear equations, then the disjunctive Rado number of the set {L₁,L₂} is the least integer n, provided that it exists, such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to either L₁ or L₂. If such an integer n does not exist, then the disjunctive Rado number is infinite. In this paper, it is shown that for all integers and b1, the disjunctive Rado number for the equations x₁+a=x₂ and x₁+b=x₂ is a+b+1-gcd(a,b) if is odd and the disjunctive Rado number for these equations is infinite otherwise. It is also shown that for all integers a>1 and b>1, the disjunctive Rado number for the equations ax₁=x₂ and bx₁=x₂ is c^s+t-1 if there exist natural numbers c,s, and t such that a=c^s and b=c^t and s+t is an odd integer and c is the largest such integer, and the disjunctive Rado number for these equations is infinite otherwise.     

Connected [a,b]-factors in Graphs

In this paper, we prove the following result: Let G be a connected graph of order n, and minimum degree . Let a and b two integers such that 2a <= b. Suppose and . Then G has a connected [a,b]-factor. Received February 10, 1998/Revised July 31, 2000     

Frobenius Problem and dead ends in integers

Let a and b be positive and relatively prime integers. We show that the following are equivalent: (i) d is a dead end in the (symmetric) Cayley graph of Z with respect to a and b, (ii) d is a Frobenius value with respect to a and b (it cannot be written as a non-negative or non-positive integer linear combination of a and b), and d is maximal (in the Cayley graph) with respect to this property. In addition, for given integers a and b, we explicitly describe all such elements in Z. Finally, we show that Z has only finitely many dead ends with respect to any finite symmetric generating set. In Appendix A we show that every finitely generated group has a generating set with respect to which dead ends exist.     

Asymptotic normality of <Emphasis Type="Italic">b</Emphasis>-additive functions on polynomial sequences in number systems

We consider b-additive functions f where b is an algebraic integer over ℤ. In particular, let p be a polynomial, then we show that the asymptotic distribution of f(⌊ p(z)⌋), where ⌊⋅⌋ denotes the integer part with respect to basis b, when z runs through the elements of the ring ℤ[b] is the normal law. This is a generalization of results of Bassily and Kátai (for the integer case) and of Gittenberger and Thuswaldner (for the Gaussian integers).     

A problem of D. H. Lehmer and its mean value

Let p be an odd prime and let a be an integer coprime to p. Denote by N (a, p) the number of pairs of integers b, c with bc ≡ a (mod p), 1 ≤ b, c < p and with b, c having different parity. The main purpose of this paper is to deduce an asymptotic formula for (N (2^sad, p) – 1/2 (p – 1)), and obtain some interesting results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positive definite quadratic forms representing integers of the form <Emphasis Type="Italic">an</Emphasis><Superscript>2</Superscript>+<Emphasis Type="Italic">b</Emphasis>

For any subset S of positive integers, a positive definite integral quadratic form is said to be S-universal if it represents every integer in the set S. In this article, we classify all binary S-universal positive definite integral quadratic forms in the case when S=S _a ={an ²∣n≥2} or S=S _a,b ={an ²+b∣n∈ℤ}, where a is a positive integer and ab is a square-free positive integer in the latter case. We also prove that there are only finitely many S _a -universal ternary quadratic forms not representing a. Finally, we show that there are exactly 15 ternary diagonal S ₁-universal quadratic forms not representing 1.     

On the diophantine equation (x − 1)(y − 1) = (z − 1)

Let k ≥ 3 be an integer. We study the possible existence of pairs of distinct positive integers (a, b) such that any of the three numbers a + 1, b + 1, and ab + 1 is a k-th power. We further investigate several related questions.