Bounds for pairs in judicious partitioning of graphs

In 2002, Bollobás and Scott posed the following problem: for an integer and a graph G of m edges, what is the smallest f (k, m ) such that V (G ) can be partitioned into V ₁,…,V _k in which for all , where denotes the number of edges with both ends in ? In this paper, we solve this problem asymptotically by showing that . We also show that V (G ) can be partitioned into such that for , where Δ denotes the maximum degree of G . This confirms a conjecture of Bollobás and Scott. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 59–70, 2017     

Brick partitions of graphs

For each rational number q=b/c where b≥c are positive integers, we define a q-brick of G to be a maximal subgraph H of G such that cH has b edge-disjoint spanning trees, and a q-superbrick of G to be a maximal subgraph H of G such that cH−e has b edge-disjoint spanning trees for all edges e of cH, where cH denotes the graph obtained from H by replacing each edge by c parallel edges. We show that the vertex sets of the q-bricks of G partition the vertex set of G, and that the vertex sets of the q-superbricks of G form a refinement of this partition. The special cases when q=1 are the partitions given by the connected components and the 2-edge-connected components of G, respectively. We obtain structural results on these partitions and describe their relationship to the principal partitions of a matroid.     

Judicious partitions of graphs

To 80^th birthday of Paul Erds     

Star partitions of graphs

Let G be a graph and n ≥ 2 an integer. We prove that the following are equivalent: (i) there is a partition (V₁,…,V_m) of V (G) such that each V_i induces one of stars K_1,1,…,K_1,n, and (ii) for every subset S of V(G), G\ S has at most n|S| components with the property that each of their blocks is an odd order complete graph. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 185–190, 1997     

Complete partitions of graphs

A complete partition of a graph G is a partition of its vertex set in which any two distinct classes are connected by an edge. Let cp(G) denote the maximum number of classes in a complete partition of G. This measure was defined in 1969 by Gupta [19], and is known to be NP-hard to compute for several classes of graphs. We obtain essentially tight lower and upper bounds on the approximability of this problem. We show that there is a randomized polynomial-time algorithm that given a graph G with n vertices, produces a complete partition of size Ω(cp(G)/√lgn). This algorithm can be derandomized. We show that the upper bound is essentially tight: there is a constant C > 1, such that if there is a randomized polynomial-time algorithm that for all large n, when given a graph G with n vertices produces a complete partition into at least C·cp(G)/√lgn classes, then NP ⊆ RTime(n ^{O(lg lg n)}). The problem of finding a complete partition of a graph is thus the first natural problem whose approximation threshold has been determined to be of the form Θ((lgn) ^c ) for some constant c strictly between 0 and 1. The work reported here is a merger of the results reported in [30] and [21].     

Skew partitions in perfect graphs

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

Star partitions on graphs

Given an undirected graph, a star partition is a partition of the nodes into subsets with at least two nodes so that the subgraph induced by each subset has a spanning star. Star partitions are related to well-known problems concerning domination in graphs and edge covering. We focus on the Constrained Star Partition Problem (CSP) that asks for finding a star partition of given cardinality. The problem is new and presents interesting peculiarities. We explore the relation between the cardinalities of star partitions and domatic bipartitions, showing that there are star partitions of any cardinality between minimum and maximum values, and that a similar but weaker result holds for domatic bipartitions. We study the computational complexity of different versions of star partition and domatic bipartition problems, proving that most of them, in particular CSP, constrained domatic bipartition and balanced domatic bipartition, are NP-complete. We also show that star partition problems are polynomial on trees and, more generally, on bounded treewidth graphs. We introduce an integer linear programming formulation that defines a polytope containing all the star partitions of a graph, showing that its vertices have only integral components for trees, which implies that linear programming can be used to solve weighted star partition problems on trees.     

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.     

Vertex partitions of r-edge-colored graphs

Let G be an edge-colored graph.The monochromatic tree partition problem is to find the minimum number of vertex disjoint monochromatic trees to cover the all vertices of G.In the authors' previous work,it has been proved that the problem is NP-complete and there does not exist any constant factor approximation algorithm for it unless P=NP.In this paper the authors show that for any fixed integer r≥5,if the edges of a graph G are colored by r colors,called an r-edge-colored graph,the problem remains NP-complete.Similar result holds for the monochromatic path(cycle)partition problem.Therefore,to find some classes of interesting graphs for which the problem can be solved in polynomial time seems interesting. A linear time algorithm for the monochromatic path partition problem for edge-colored trees is given.     

Vertex partitions of chordal graphs

A k‐tree is a chordal graph with no (k + 2)‐clique. An ?‐tree‐partition of a graph G is a vertex partition of G into ‘bags,’ such that contracting each bag to a single vertex gives an ?‐tree (after deleting loops and replacing parallel edges by a single edge). We prove that for all k ≥ ? ≥ 0, every k‐tree has an ?‐tree‐partition in which each bag induces a connected ‐tree. An analogous result is proved for oriented k‐trees. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 167–172, 2006     

Judicious partitions of directed graphs

The area of judicious partitioning considers the general family of partitioning problems in which one seeks to optimize several parameters simultaneously, and these problems have been widely studied in various combinatorial contexts. In this paper, we study essentially the most fundamental judicious partitioning problem for directed graphs, which naturally extends the classical Max Cut problem to this setting: we seek bipartitions in which many edges cross in each direction. It is easy to see that a minimum outdegree condition is required in order for the problem to be nontrivial, and we prove that every directed graph with m edges and minimum outdegree at least two admits a bipartition in which at least edges cross in each direction. We also prove that if the minimum outdegree is at least three, then the constant can be increased to . If the minimum outdegree tends to infinity with n, then the constant increases to . All of these constants are best‐possible, and provide asymptotic answers to a question of Alex Scott. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 147–170, 2016     

Domatic partitions of computable graphs

Given a graph G, we say that a subset D of the vertex set V is a dominating set if it is near all the vertices, in that every vertex outside of D is adjacent to a vertex in D. A domatic k-partition of G is a partition of V into k dominating sets. In this paper, we will consider issues of computability related to domatic partitions of computable graphs. Our investigation will center on answering two types of questions for the case when k = 3. First, if domatic 3-partitions exist in a computable graph, how complicated can they be? Second, a decision problem: given a graph, how difficult is it to decide whether it has a domatic 3-partition? We will completely classify this decision problem for highly computable graphs, locally finite computable graphs, and computable graphs in general. Specifically, we show the decision problems for these kinds of graphs to be ${\Pi^{0}_{1}}$ -, ${\Pi^{0}_{2}}$ -, and ${\Sigma^{1}_{1}}$ -complete, respectively.     

Bounds for point recolouring in geometric graphs

We examine a recolouring scheme ostensibly used to assist in classifying geographic data. Given a drawing of a graph with bi-chromatic points, where the points are the vertices of the graph, a point can be recoloured if it is surrounded by neighbours of the opposite colour. The notion of surrounded is defined as a contiguous subset of neighbours that span an angle greater than 180 degrees. The recolouring of surrounded points continues in sequence, in no particular order, until no point remains surrounded. We show that for some classes of graphs the process terminates in a polynomial number of steps. On the other hand, there are classes of graphs with infinite recolouring sequences.     

Bounds for algebraic gossip on graphs

We study the stopping times of gossip algorithms for network coding. We analyze algebraic gossip (i.e., random linear coding) and consider three gossip algorithms for information spreading: Pull, Push, and Exchange. The stopping time of algebraic gossip is known to be linear for the complete graph, but the question of determining a tight upper bound or lower bounds for general graphs is still open. We take a major step in solving this question, and prove that algebraic gossip on any graph of size n is O(Δn) where Δ is the maximum degree of the graph. This leads to a tight bound of for bounded degree graphs and an upper bound of O(n²) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of . Our proofs use a novel method that relies on Jackson's queuing theorem to analyze the stopping time of network coding; this technique is likely to become useful for future research. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 185–217, 2014     

On isomorphic linear partitions in cubic graphs

A linear forest is a graph whose connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. It is well known that la(G)=2 when G is a cubic graph and Wormald [N. Wormald, Problem 13, Ars Combinatoria 23(A) (1987) 332-334] conjectured that if |V(G)|≡0 (mod 4), then it is always possible to find a linear partition in two isomorphic linear forests. Here, we give some new results concerning this conjecture.     

Vertex partitions of non-complete graphs into connected monochromatic k-regular graphs

In a landmark paper, Erd?s et al. (1991) [3] proved that if $G$ is a complete graph whose edges are colored with $r$ colors then the vertex set of $G$ can be partitioned into at most $c{r}^{2}logr$ monochromatic, vertex disjoint cycles for some constant $c$. Sárközy extended this result to non-complete graphs, and Sárközy and Selkow extended it to $k$-regular subgraphs. Generalizing these two results, we show that if $G$ is a graph with independence number $\alpha \left(G\right)=\alpha$ whose edges are colored with $r$ colors then the vertex set of $G$ can be partitioned into at most ${\left(\alpha r\right)}^{c(\alpha rlog\left(\alpha r\right)+k)}$ vertex disjoint connected monochromatic$k$-regular subgraphs of $G$ for some constant $c$.