# Vol. 7

### 1. Infinite families of accelerated series for some classical constants by the Markov-WZ Method

In this article we show the Markov-WZ Method in action as it finds rapidly converging series representations for a given hypergeometric series. We demonstrate the method by finding new representations for log(2),ζ (2) and ζ (3).

### 2. Graphs of low chordality

The chordality of a graph with at least one cycle is the length of the longest induced cycle in it. The odd (even) chordality is defined to be the length of the longest induced odd (even) cycle in it. Chordal graphs have chordality at most 3. We show that co-circular-arc graphs and co-circle graphs have even chordality at most 4. We also identify few other classes of graphs having bounded (by a constant) chordality values.

### 3. Acyclic, Star and Oriented Colourings of Graph Subdivisions

Let G be a graph with chromatic number χ (G). A vertex colouring of G is \emphacyclic if each bichromatic subgraph is a forest. A \emphstar colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χ _a(G) and χ _s(G) denote the acyclic and star chromatic numbers of G. This paper investigates acyclic and star colourings of subdivisions. Let G' be the graph obtained from G by subdividing each edge once. We prove that acyclic (respectively, star) colourings of G' correspond to vertex partitions of G in which each subgraph has small arboricity (chromatic index). It follows that χ _a(G'), χ _s(G') and χ (G) are tied, in the sense that each is bounded by a function of the other. Moreover the binding functions that we establish are all tight. The \emphoriented chromatic number χ ^→(G) of an (undirected) graph G is the maximum, taken over all orientations D of G, of the minimum number of colours in a vertex colouring of D such that between any two colour classes, all edges have the same direction. We prove that χ ^→(G')=χ (G) whenever χ (G)≥ 9.

### 4. Recognizing Maximal Unfrozen Graphs with respect to Independent Sets is CO-NP-complete

A graph is unfrozen with respect to k independent set if it has an independent set of size k after the addition of any edge. The problem of recognizing such graphs is known to be NP-complete. A graph is maximal if the addition of one edge means it is no longer unfrozen. We designate the problem of recognizing maximal unfrozen graphs as MAX(U(k-SET)) and show that this problem is CO-NP-complete. This partially fills a gap in known complexity cases of maximal NP-complete problems, and raises some interesting open conjectures discussed in the conclusion.

### 5. Stacks, Queues and Tracks: Layouts of Graph Subdivisions

A \emphk-stack layout (respectively, \emphk-queuelayout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing (non-nested) edges with respect to the vertex ordering. A \emphk-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The \emphstack-number (respectively, \emphqueue-number, \emphtrack-number) of a graph G, denoted by sn(G) (qn(G), tn(G)), is the minimum k such that G has a k-stack (k-queue, k-track) layout.\par This paper studies stack, queue, and track layouts of graph subdivisions. It is known that every graph has a 3-stack subdivision. The best known upper bound on the number of division vertices per edge in a 3-stack subdivision of an n-vertex graph G is improved from O(log n) to O(log min\sn(G),qn(G)\). This result reduces the question of whether queue-number is bounded by stack-number to whether 3-stack graphs have bounded queue number.\par It is proved that every graph has a 2-queue subdivision, a 4-track subdivision, and a mixed 1-stack 1-queue subdivision. All these values are optimal for every non-planar graph. In addition, we characterise those graphs with k-stack, k-queue, and k-track subdivisions, for all values of k. The number of division vertices per edge in the case of 2-queue and 4-track subdivisions, namely O(log qn(G)), is optimal to within a constant factor, for every graph […]

### 6. Tilings from some non-irreducible, Pisot substitutions

A generating method of self-affine tilings for Pisot, unimodular, irreducible substitutions, as well as the fact that the associated substitution dynamical systems are isomorphic to rotations on the torus are established in P. Arnoux and S. Ito. The aim of this paper is to extend these facts in the case where the characteristic polynomial of a substitution is non-irreducible for a special class of substitutions on five letters. Finally we show that the substitution dynamical systems for this class are isomorphic to induced transformations of rotations on the torus.

### 7. Dihedral f-tilings of the sphere by rhombi and triangles

We classify, up to an isomorphism, the class of all dihedral f-tilings of S^2, whose prototiles are a spherical triangle and a spherical rhombus. The equiangular case was considered and classified in Ana M. Breda and Altino F. Santos, Dihedral f-tilings of the sphere by spherical triangles and equiangular well-centered quadrangles. Here we complete the classification considering the case of non-equiangular rhombi.

### 8. On the maximum average degree and the incidence chromatic number of a graph

We prove that the incidence chromatic number of every 3-degenerated graph G is at most Δ (G)+4. It is known that the incidence chromatic number of every graph G with maximum average degree mad(G)<3 is at most Δ (G)+3. We show that when Δ (G) ≥ 5, this bound may be decreased to Δ (G)+2. Moreover, we show that for every graph G with mad(G)<22/9 (resp. with mad(G)<16/7 and Δ (G)≥ 4), this bound may be decreased to Δ (G)+2 (resp. to Δ (G)+1).

### 9. Karp-Miller Trees for a Branching Extension of VASS

We study BVASS (Branching VASS) which extend VASS (Vector Addition Systems with States) by allowing addition transitions that merge two configurations. Runs in BVASS are tree-like structures instead of linear ones as for VASS. We show that the construction of Karp-Miller trees for VASS can be extended to BVASS. This entails that the coverability set for BVASS is computable. This allows us to obtain decidability results for certain classes of equational tree automata with an associative-commutative symbol. Recent independent work by de Groote et al. implies that decidability of reachability in BVASS is equivalent to decidability of provability in MELL (multiplicative exponential linear logic), which is still an open problem. Hence our results are also a step towards answering this question in the affirmative.

### 10. On the Recognition of Bipolarizable and P_4-simplicial Graphs

The classes of Raspail (also known as Bipolarizable) and P_4-simplicial graphs were introduced by Hoàng and Reed who showed that both classes are perfectly orderable and admit polynomial-time recognition algorithms HR1. In this paper, we consider the recognition problem on these classes of graphs and present algorithms that solve it in O(n m) time. In particular, we prove properties and show that we can produce bipolarizable and P_4-simplicial orderings on the vertices of the input graph G, if such orderings exist, working only on P_3s that participate in a P_4 of G. The proposed recognition algorithms are simple, use simple data structures and both require O(n + m) space. Additionally, we show how our recognition algorithms can be augmented to provide certificates, whenever they decide that G is not bipolarizable or P_4-simplicial; the augmentation takes O(n + m) time and space. Finally, we include a diagram on class inclusions and the currently best recognition time complexities for a number of perfectly orderable classes of graphs.

### 11. Queue Layouts of Graph Products and Powers

A \emphk-queue layout of a graph G consists of a linear order σ of V(G), and a partition of E(G) into k sets, each of which contains no two edges that are nested in σ . This paper studies queue layouts of graph products and powers

### 12. Connectedness of number theoretical tilings

Let T=T(A,D) be a self-affine tile in \reals^n defined by an integral expanding matrix A and a digit set D. In connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers \0,1,..., |det(A)|-1\. It is shown that in \reals^3 and \reals^4, for any integral expanding matrix A, T(A,D) is connected. We also study the connectedness of Pisot dual tilings which play an important role in the study of β -expansion, substitution and symbolic dynamical system. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. This is naturally expected since the digit set consists of consecutive integers as above. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, a complete classification of the β -expansion of 1 for quartic Pisot units is given.

### 13. Enumeration of Binary Trees and Universal Types

Binary unlabeled ordered trees (further called binary trees) were studied at least since Euler, who enumerated them. The number of such trees with n nodes is now known as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g., height and path length distributions for a randomly selected tree). Binary trees find an abundance of applications in computer science. However, recently Seroussi posed a new and interesting problem motivated by information theory considerations: how many binary trees of a \emphgiven path length (sum of depths) are there? This question arose in the study of \emphuniversal types of sequences. Two sequences of length p have the same universal type if they generate the same set of phrases in the incremental parsing of the Lempel-Ziv'78 scheme since one proves that such sequences converge to the same empirical distribution. It turns out that the number of distinct types of sequences of length p corresponds to the number of binary (unlabeled and ordered) trees, T_p, of given path length p (and also the number of distinct Lempel-Ziv'78 parsings of length p sequences). We first show that the number of binary trees with given path length p is asymptotically equal to T_p ~ 2^2p/(log_2 p)(1+O(log ^-2/3 p)). Then we establish various limiting distributions for the number of nodes (number of phrases in the Lempel-Ziv'78 scheme) when a tree is selected randomly among all trees of given path length […]

### 14. Two Pile Move-Size Dynamic Nim

The purpose of this paper is to solve a special class of combinational games consisting of two-pile counter pickup games for which the maximum number of counters that can be removed on each successive move changes during the play of the games. Two players alternate moving. Each player in his turn first chooses one of the piles, and his choice of piles can change from move to move. He then removes counters from this chosen pile. A function f:Z^+ → Z^+ is given which determines the maximum size of the next move in terms of the current move size. The game ends as soon as one of the two piles is empty, and the winner is the last player to move in the game. The games for which f(k)=k, f(k)=2k, and f(k)=3k use the same formula for computing the smallest winning move size. Here we find all the functions f for which this formula works, and we also give the winning strategy for each function. See Holshouser, A, James Rudzinski and Harold Reiter: Dynamic One-Pile Nim for a discussion of the single pile game.

### 15. Some equinumerous pattern-avoiding classes of permutations

Suppose that p,q,r,s are non-negative integers with m=p+q+r+s. The class X(p,q,r,s) of permutations that contain no pattern of the form α β γ where |α |=r, |γ |=s and β is any arrangement of \1,2,\ldots,p\∪ \m-q+1, m-q+2, \ldots,m\ is considered. A recurrence relation to enumerate the permutations of X(p,q,r,s) is established. The method of proof also shows that X(p,q,r,s)=X(p,q,1,0)X(1,0,r,s) in the sense of permutational composition.\par 2000 MATHEMATICS SUBJECT CLASSIFICATION: 05A05

### 16. An extremal problem on potentially K_p,1,1-graphic sequences

A sequence S is potentially K_p,1,1 graphical if it has a realization containing a K_p,1,1 as a subgraph, where K_p,1,1 is a complete 3-partite graph with partition sizes p,1,1. Let σ (K_p,1,1, n) denote the smallest degree sum such that every n-term graphical sequence S with σ (S)≥ σ (K_p,1,1, n) is potentially K_p,1,1 graphical. In this paper, we prove that σ (K_p,1,1, n)≥ 2[((p+1)(n-1)+2)/2] for n ≥ p+2. We conjecture that equality holds for n ≥ 2p+4. We prove that this conjecture is true for p = 3. AMS Subject Classifications: 05C07, 05C35

### 17. Algebraic Elimination of epsilon-transitions

We present here algebraic formulas associating a k-automaton to a k-epsilon-automaton. The existence depends on the definition of the star of matrices and of elements in the semiring k. For this reason, we present the theorem which allows the transformation of k-epsilon-automata into k-automata. The two automata have the same behaviour.