Infinite partitions of random graphs

We determine the set of canonical equivalence relations on [G]ⁿ, where G is a random graph, extending the result of Erd?s and Rado for the integers to random graphs.     

Skew partitions in perfect graphs

We discuss some new and old results about skew partitions in perfect graphs.     

Broadcasting secure messages via optimal independent spanning trees in folded hypercubes

Fault-tolerant broadcasting and secure message distribution are important issues for network applications. It is a common idea to design multiple spanning trees with a specific property in the underlying graph of a network to serve as a broadcasting scheme or a distribution protocol for receiving high levels of fault-tolerance and security. An n-dimensional folded hypercube, denoted by FQ_n, is a strengthening variation of hypercube by adding additional links between nodes that have the furthest Hamming distance. In, [12], Ho(1990) proposed an algorithm for constructing n+1 edge-disjoint spanning trees each with a height twice the diameter of FQ_n. Yang et al. (2009), [29] recently proved that Ho’s spanning trees are indeed independent, i.e., any two spanning trees have the same root, say r, and for any other node v≠r, the two different paths from v to r, one path in each tree, are internally node-disjoint. In this paper, we provide another construction scheme to produce n+1 independent spanning trees of FQ_n, where the height of each tree is equal to the diameter of FQ_n plus one. As a result, the heights of independent spanning trees constructed in this paper are shown to be optimal.     

Ordered partitions and drawings of rooted plane trees

We study the bounded regions in a generic slice of the hyperplane arrangement in Rⁿ${\mathbb{R}}^n$ consisting of the hyperplanes defined by _xi$x_i$ and _xi+_xj$x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial objects, including the ordered partitions of [n]$\left[n\right]$ all of whose left-to-right minima occur at odd locations and the drawings of rooted plane trees with n+1$n+1$ vertices. These are sequences of rooted plane trees such that each tree in a sequence can be obtained from the next one by removing a leaf.     

Clique partitions of complements of forests and bounded degree graphs

In this paper, we prove that for any forest F⊂K_n, the edges of E(K_n)?E(F) can be partitioned into O(nlogn) cliques. This extends earlier results on clique partitions of the complement of a perfect matching and of a hamiltonian path in K_n.In the second part of the paper, we show that for n sufficiently large and any ε∈(0,1], if a graph G has maximum degree O(n^1-ε), then the edges of E(K_n)?E(G) can be partitioned into cliques provided there exist certain Steiner systems. Furthermore, we show that there are such graphs G for which Ω(ε²n^2-2ε) cliques are required in every clique partition of E(K_n)?E(G).     

Random partitions and edge-disjoint Hamiltonian cycles

Nash-Williams [1] proved that every graph with n   vertices and minimum degree n/2$n/2$ has at least ⌊5n/224⌋$\lfloor 5n/224\rfloor$ edge-disjoint Hamiltonian cycles. In [2], he raised the question of determining the maximum number of edge-disjoint Hamiltonian cycles, showing an upper bound of ⌊(n+4)/8⌋$\lfloor (n+4)/8\rfloor$.     

Idomatic partitions of direct products of complete graphs

In this paper we give a full characterization of the idomatic partitions of the direct product of three complete graphs. We also show how to use such a characterization in order to construct idomatic partitions of the direct product of finitely many complete graphs.     

Double perfect partitions

MacMahon [Combinatory Analysis, vols. I and II, Cambridge University Press, Cambridge, 1915, 1916 (reprinted, Chelsea, 1960)] introduced a perfect partition of positive integer n, which is a partition such that every positive integer less than or equal to n can be uniquely represented by the sum of its parts. We generalize perfect partition and find a relation with ordered factorizations.     

Edge partitions of optimal 2-plane and 3-plane graphs

A topological graph is a graph drawn in the plane. A topological graph is $k$-plane, $k>0$, if each edge is crossed at most $k$ times. We study the problem of partitioning the edges of a $k$-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for $k=1$, we focus on optimal 2-plane and on optimal 3-plane graphs, which are 2-plane and 3-plane graphs with maximum density. We prove the following results. (i) It is not possible to partition the edges of a simple (i.e., with neither self-loops nor parallel edges) optimal 2-plane graph into a 1-plane graph and a forest, while (ii) an edge partition formed by a 1-plane graph and two plane forests always exists and can be computed in linear time. (iii) There exist efficient algorithms to partition the edges of a simple optimal 2-plane graph into a 1-plane graph and a plane graph with maximum vertex degree at most 12, or with maximum vertex degree at most 8 if the optimal2-plane graph is such that its crossing-free edges form a graph with no separating triangles. (iv) There exists an infinite family of simple optimal 2-plane graphs such that in any edge partition composed of a 1-plane graph and a plane graph, the plane graph has maximum vertex degree at least 6 and the 1-plane graph has maximum vertex degree at least 12. (v) Every optimal 3-plane graph whose crossing-free edges form a biconnected graph can be decomposed, in linear time, into a 2-plane graph and two plane forests.     

Data structures for maintaining set partitions

Efficiently maintaining the partition induced by a set of features is an important problem in building decision‐tree classifiers. In order to identify a small set of discriminating features, we need the capability of efficiently adding and removing specific features and determining the effect of these changes on the induced classification or partition. In this paper we introduce a variety of randomized and deterministic data structures to support these operations on both general and geometrically induced set partitions. We give both Monte Carlo and Las Vegas data structures that realize near‐optimal time bounds and are practical to implement. We then provide a faster solution to this problem in the geometric setting. Finally, we present a data structure that efficiently estimates the number of partitions separating elements. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

The threshold for combs in random graphs

For let denote the tree consisting of an ‐vertex path with disjoint ‐vertex paths beginning at each of its vertices. An old conjecture says that for any the threshold for the random graph to contain is at . Here we verify this for with any fixed . In a companion paper, using very different methods, we treat the complementary range, proving the conjecture for (with ). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 794–802, 2016     

Critical random graphs and the structure of a minimum spanning tree

We consider the complete graph on n vertices whose edges are weighted by independent and identically distributed edge weights and build the associated minimum weight spanning tree. We show that if the random weights are all distinct, then the expected diameter of such a tree is Θ(n^1/3). This settles a question of Frieze and Mc‐Diarmid (Random Struct Algorithm 10 (1997), 5–42). The proofs are based on a precise analysis of the behavior of random graphs around the critical point. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Rainbow spanning trees in complete graphs colored by one‐factorizations

Brualdi and Hollingsworth conjectured that, for even n, in a proper edge coloring of using precisely colors, the edge set can be partitioned into spanning trees which are rainbow (and hence, precisely one edge from each color class is in each spanning tree). They proved that there always are two edge disjoint rainbow spanning trees. Krussel, Marshall, and Verrall improved this to three edge disjoint rainbow spanning trees. Recently, Carraher, Hartke and the author proved a theorem improving this to rainbow spanning trees, even when more general edge colorings of are considered. In this article, we show that if is properly edge colored with colors, a positive fraction of the edges can be covered by edge disjoint rainbow spanning trees.     

Counting degree sequences of spanning trees in bipartite graphs: A graph-theoretic proof

Given a bipartite graph with bipartition each spanning tree in has a degree sequence on and one on . Löhne and Rudloff showed that the number of possible degree sequences on equals the number of possible degree sequences on . Their proof uses a non-trivial characterization of degree sequences by -draconian sequences based on polyhedral results of Postnikov. In this paper, we give a purely graph-theoretic proof of their result.     

Rainbow structures in locally bounded colorings of graphs

We study approximate decompositions of edge‐colored quasirandom graphs into rainbow spanning structures: an edge‐coloring of a graph is locally ‐bounded if every vertex is incident to at most edges of each color, and is (globally) ‐bounded if every color appears at most times. Our results imply the existence of: (1) approximate decompositions of properly edge‐colored into rainbow almost‐spanning cycles; (2) approximate decompositions of edge‐colored into rainbow Hamilton cycles, provided that the coloring is ‐bounded and locally ‐bounded; and (3) an approximate decomposition into full transversals of any array, provided each symbol appears times in total and only times in each row or column. Apart from the logarithmic factors, these bounds are essentially best possible. We also prove analogues for rainbow ‐factors, where is any fixed graph. Both (1) and (2) imply approximate versions of the Brualdi‐Hollingsworth conjecture on decompositions into rainbow spanning trees.     

Polychromatic colorings of rectangular partitions

A rectangular partition is a partition of a plane rectangle into an arbitrary number of non-overlapping rectangles such that no four rectangles share a corner. In this note, it is proven that every rectangular partition admits a vertex coloring with four colors such that every rectangle, except possibly the outer rectangle, has all four colors on its boundary. This settles a conjecture of Dinitz et al. [Y. Dinitz, M.J. Katz, R. Krakovski, Guarding rectangular partitions, in: Abstracts 23rd Euro. Workshop Comput. Geom., 2007, pp. 30-33]. The proof is short, simple and based on 4-edge-colorability of a specific class of planar graphs.     

Nodal domains and spectral minimal partitions

We consider two-dimensional Schrödinger operators in bounded domains. We analyze relations between the nodal domains of eigenfunctions, spectral minimal partitions and spectral properties of the corresponding operator. The main results concern the existence and regularity of the minimal partitions and the characterization of the minimal partitions associated with nodal sets as the nodal domains of Courant-sharp eigenfunctions.