# Vol. 13 no. 3

### 1. The largest singletons in weighted set partitions and its applications

Recently, Deutsch and Elizalde studied the largest fixed points of permutations. Motivated by their work, we consider the analogous problems in weighted set partitions. Let A (n,k) (t) denote the total weight of partitions on [n + 1] = \1,2,..., n + 1\ with the largest singleton \k + 1\. In this paper, explicit formulas for A (n,k) (t) and many combinatorial identities involving A (n,k) (t) are obtained by umbral operators and combinatorial methods. In particular, the permutation case leads to an identity related to tree enumerations, namely, [GRAPHICS] where D-k is the number of permutations of [k] with no fixed points.
Section: Combinatorics

### 2. Digital search trees with m trees: Level polynomials and insertion costs

We adapt a novel idea of Cichon's related to Approximate Counting to the present instance of Digital Search Trees, by using m (instead of one) such trees. We investigate the level polynomials, which have as coefficients the expected numbers of data on a given level, and the insertion costs. The level polynomials can be precisely described, thanks to formulae from q-analysis. The asymptotics of expectation and variance of the insertion cost are fairly standard these days and done with Rice's method.
Section: Analysis of Algorithms

### 3. On the number of maximal independent sets in a graph

Miller and Muller (1960) and independently Moon and Moser (1965) determined the maximum number of maximal independent sets in an n-vertex graph. We give a new and simple proof of this result.
Section: Combinatorics

### 4. New Upper Bounds for the Heights of Some Light Subgraphs in 1-Planar Graphs with High Minimum Degree

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph of minimum degree 6 contains a copy of 4-cycle with all vertices of degree at most 19. In addition, we also show that the complete graph K 4 is light in the family of 1-planar graphs of minimum degree 7, with its height at most 11.
Section: Combinatorics

### 5. The Variance of the Profile in Digital Search Trees

What today we call digital search tree (DST) is Coffman and Eve's sequence tree introduced in 1970. A digital search tree is a binary tree whose ordering of nodes is based on the values of bits in the binary representation of a node's key. In fact, a digital search tree is a digital tree in which strings (keys, words) are stored directly in internal nodes. The profile of a digital search tree is a parameter that counts the number of nodes at the same distance from the root. In this paper we concentrate on external profile, i.e., the number of external nodes at level k when n strings are sorted. By assuming that the n input strings are independent and follow a (binary) memoryless source the asymptotic behaviour of the average profile was determined by Drmota and Szpankowski (2011). The purpose of the present paper is to extend their analysis and to provide a precise analysis of variance of the profile. The main (technical) difference is that we have to deal with an inhomogeneous part in a proper functional-differential equations satisfied by the second moment and Poisson variance. However, we show that the variance is asymptotically of the same order as the expected value which implies concentration. These results are derived by methods of analytic combinatorics such as generating functions, Mellin transform, Poissonization, the saddle point method and singularity analysis.
Section: Analysis of Algorithms

### 6. On the Book Thickness of k-Trees

Every k-tree has book thickness at most k + 1, and this bound is best possible for all k \textgreater= 3. Vandenbussche et al. [SIAM J. Discrete Math., 2009] proved that every k-tree that has a smooth degree-3 tree decomposition with width k has book thickness at most k. We prove this result is best possible for k \textgreater= 4, by constructing a k-tree with book thickness k + 1 that has a smooth degree-4 tree decomposition with width k. This solves an open problem of Vandenbussche et al.
Section: Graph and Algorithms

### 7. On the complexity of vertex-coloring edge-weightings

Given a graph G = (V; E) and a weight function omega : E -\textgreater R, a coloring of vertices of G, induced by omega, is defined by chi(omega) (nu) = Sigma(e(sic)nu) omega (e) for all nu is an element of V. In this paper, we show that determining whether a particular graph has a weighting of the edges from \1, 2\ that induces a proper vertex coloring is NP-complete.
Section: Graph and Algorithms

### 8. On the number of factors in codings of three interval exchange

We consider exchange of three intervals with permutation (3, 2, 1). The aim of this paper is to count the cardinality of the set 3iet (N) of all words of length N which appear as factors in infinite words coding such transformations. We use the strong relation of 3iet words and words coding exchange of two intervals, i.e., Sturmian words. The known asymptotic formula #2iet(N)/N-3 similar to 1/pi(2) for the number of Sturmian factors allows us to find bounds 1/3 pi(2) +o(1) \textless= #3iet(N)N-4 \textless= 2 pi(2) + o(1)
Section: Graph Theory

### 9. On packing of two copies of a hypergraph

A 2-packing of a hypergraph H is a permutation sigma on V (H) such that if an edge e belongs to epsilon(H), then sigma(e) does not belong to epsilon(H). Let H be a hypergraph of order n which contains edges of cardinality at least 2 and at most n - 2. We prove that if H has at most n - 2 edges then it is 2-packable.
Section: Graph and Algorithms

### 10. Nordhaus-Gaddum Type Results for Total Domination

A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper we study Nordhaus-Gaddum-type results for total domination. We examine the sum and product of γt(G1) and γt(G2) where G1 ⊕G2 = K(s,s), and γt is the total domination number. We show that the maximum value of the sum of the total domination numbers of G1 and G2 is 2s+4, with equality if and only if G1 = sK2 or G2 = sK2, while the maximum value of the product of the total domination numbers of G1 and G2 is max{8s,⌊(s+6)2/4 ⌋}.
Section: Graph Theory

### 11. 8-star-choosability of a graph with maximum average degree less than 3

A proper vertex coloring of a graphGis called a star-coloring if there is no path on four vertices assigned to two colors. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring c such that c(v) epsilon L(v). If G is L-star-colorable for any list assignment L with vertical bar L(v)vertical bar \textgreater= k for all v epsilon V(G), then G is called k-star-choosable. The star list chromatic number of G, denoted by X-s(l)(G), is the smallest integer k such that G is k-star-choosable. In this article, we prove that every graph G with maximum average degree less than 3 is 8-star-choosable. This extends a result that planar graphs of girth at least 6 are 8-star-choosable [A. Kundgen, C. Timmons, Star coloring planar graphs from small lists, J. Graph Theory, 63(4): 324-337, 2010].
Section: Graph and Algorithms