Partitions of graphs into small and large sets

Let $G$ be a graph on $n$ vertices. We call a subset $A$ of the vertex set $V(G)$ \emph{$k$-small} if, for every vertex $v \in A$, $\deg(v) \le n - |A| + k$. A subset $B \subseteq V(G)$ is called \emph{$k$-large} if, for every vertex $u \in B$, $\deg(u) \ge |B| - k - 1$. Moreover, we denote by $\varphi_k(G)$ the minimum integer $t$ such that there is a partition of $V(G)$ into $t$ $k$-small sets, and by $\Omega_k(G)$ the minimum integer $t$ such that there is a partition of $V(G)$ into $t$ $k$-large sets. In this paper, we will show tight connections between $k$-small sets, respectively $k$-large sets, and the $k$-independence number, the clique number and the chromatic number of a graph. We shall develop greedy algorithms to compute in linear time both $\varphi_k(G)$ and $\Omega_k(G)$ and prove various sharp inequalities concerning these parameters, which we will use to obtain refinements of the Caro-Wei Theorem, the Tur\'an Theorem and the Hansen-Zheng Theorem among other things.Comment: 21 page

k-Efficient Partitions of Graphs

A set S = {u1, u2,..., ut} of vertices of G is an efficient dominating set if every vertex of G is dominated exactly once by the vertices of S. Letting Ui denote the set of vertices dominated by ui, we note that {U1, U2,... Ut} is a partition of the vertex set of G and that each Ui contains the vertex ui and all the vertices at distance 1 from it in G. In this paper, we generalize the concept of efficient domination by considering k-efficient domination partitions of the vertex set of G, where each element of the partition is a set consisting of a vertex ui and all the vertices at distance di from it, where di ∈ {0, 1,..., k}. For any integer k ≥ 0, the k-efficient domination number of G equals the minimum order of a k-efficient partition of G. We determine bounds on the k-efficient domination number for general graphs, and for k ∈ {1, 2}, we give exact values for some graph families. Complexity results are also obtained

Counting Connected Partitions of Graphs

Motivated by the theorem of Gy\H ori and Lov\'asz, we consider the following problem. For a connected graph $G$ on $n$ vertices and $m$ edges determine the number $P(G,k)$ of unordered solutions of positive integers $\sum_{i=1}^k m_i = m$ such that every $m_i$ is realized by a connected subgraph $H_i$ of $G$ with $m_i$ edges such that $\cup_{i=1}^kE(H_i)=E(G)$. We also consider the vertex-partition analogue. We prove various lower bounds on $P(G,k)$ as a function of the number $n$ of vertices in $G$, as a function of the average degree $d$ of $G$, and also as the size $\mathrm{CMC}_r(G)$ of $r$-partite connected maximum cuts of $G$. Those three lower bounds are tight up to a multiplicative constant. We also prove that the number $\pi(G,k)$ of unordered $k$-tuples with $\sum_{i=1}^kn_i=n$, that are realizable by vertex partitions into $k$ connected parts of respective sizes $n_1,n_2,\dots,n_k$, is $\Omega(d^{k-1})$

On tree-partitions of graphs

A graph G admits a tree-partition of width k if its vertex set can be partitioned into sets of size at most k so that the graph obtained by identifying the vertices in each set of the partition, and then deleting loops and parallel edges, is a forest. In the paper, we characterize the classes of graphs (finite and infinite) of bounded tree-partition-width in terms of excluded topological minors

A characterization and an application of weight-regular partitions of graphs

A natural generalization of a regular (or equitable) partition of a graph, which makes sense also for non-regular graphs, is the so-called weight-regular partition, which gives to each vertex $u\in V$ a weight that equals the corresponding entry $\nu_u$ of the Perron eigenvector $\mathbf{\nu}$. This paper contains three main results related to weight-regular partitions of a graph. The first is a characterization of weight-regular partitions in terms of double stochastic matrices. Inspired by a characterization of regular graphs by Hoffman, we also provide a new characterization of weight-regularity by using a Hoffman-like polynomial. As a corollary, we obtain Hoffman's result for regular graphs. In addition, we show an application of weight-regular partitions to study graphs that attain equality in the classical Hoffman's lower bound for the chromatic number of a graph, and we show that weight-regularity provides a condition under which Hoffman's bound can be improved

Judicious partitions of graphs and hypergraphs

Classical partitioning problems, like the Max-Cut problem, ask for partitions that optimize one quantity, which are important to such fields as VLSI design, combinatorial optimization, and computer science. Judicious partitioning problems on graphs or hypergraphs ask for partitions that optimize several quantities simultaneously. In this dissertation, we work on judicious partitions of graphs and hypergraphs, and solve or asymptotically solve several open problems of Bollobas and Scott on judicious partitions, using the probabilistic method and extremal techniques.Ph.D.Committee Chair: Yu, Xingxing; Committee Member: Shapira, Asaf; Committee Member: Tetali, Prasad; Committee Member: Thomas, Robin; Committee Member: Vigoda, Eri