Judicious partitions of graphs
نویسندگان
چکیده
The problem of finding good lower bounds on the size of the largest bipartite subgraph of a given graph has received a fair amount of attention. In particular, improving a result of Erdős ([10]; see also [11] for related problems), Edwards [9] proved the essentially best possible assertion that every graph with n vertices and m edges has a bipartite subgraph with at least m/2 + (n − 1)/4 edges. More recently, Andersen et al [1] and Erdős et al [12] gave lower bounds for the size of the largest k-partite subgraph of a given graph, and Shearer [18] and Ngoc and Tuza [15] gave bounds for the lowest bipartite subgraph of a triangle free graph. Various algorithms for finding large k-partite subgraphs have been considered in [16], [17] and [15]. In this paper we consider a naturally related question. Given a graph G, we again consider partitions V1, . . . , Vk of V (G) into k sets. We ask, however, for the minimal value of max1≤i≤k e(G[Vi]). Thus we seek a partition of V (G) in which every class induces relatively few edges, in contrast to the problem of finding the largest k-partite subgraph of a given graph G, which asks for a partition in which the total number of edges induced by the classes is small. As we shall see, the nature of the problem depends on the size of the graph. Our first aim in this paper is to prove a bound valid for all graphs: although the bound is best possible, the only graphs on which it is attained are very small. Our other aim is to prove a much better and essentially best possible bound for graphs with many edges. In §1 we shall give our exact result: for any k, and for any graph G, there is a partition V (G) = ⋃k i=1 Vi such that e(G[Vi]) ≤ e(G)/ ( k+1 2 ) for i = 1, . . . , k. For a given value of k. this inequality is best possible. The improvement for graphs with many edges will be given in §2: as we shall see, the upper bound can almost be halved. In fact, we can even demand
منابع مشابه
Judicious k-partitions of graphs
Judicious partition problems ask for partitions of the vertex set of graphs so that several quantities are optimized simultaneously. In this paper, we answer the following judicious partition question of Bollobás and Scott [6] in the affirmative: For any positive integer k and for any graph G of size m, does there exist a partition of V (G) into V1, . . . , Vk such that the total number of edge...
متن کاملk-Efficient partitions of graphs
A set $S = {u_1,u_2, ldots, u_t}$ of vertices of $G$ is an efficientdominating set if every vertex of $G$ is dominated exactly once by thevertices of $S$. Letting $U_i$ denote the set of vertices dominated by $u_i$%, we note that ${U_1, U_2, ldots U_t}$ is a partition of the vertex setof $G$ and that each $U_i$ contains the vertex $u_i$ and all the vertices atdistance~1 from it in $G$. In this ...
متن کاملJudicious partitions of bounded-degree graphs
We prove results on partitioning graphs G with bounded maximum degree. In particular, we provide optimal bounds for bipartitions V (G) = V1 ∪ V2 in which we minimize max{e(V1), e(V2)}.
متن کاملProblems and results on judicious partitions
We present a few results and a larger number of questions concerning partitions of graphs or hypergraphs, where the objective is to maximize or minimize several quantities simultaneously. We consider a variety of extremal problems; many of these also have algorithmic counterparts.
متن کاملExact Bounds for Judicious Partitions of Graphs
Edwards showed that every graph of size m ≥ 1 has a bipartite subgraph of size at least m/2 + √ m/8 + 1/64− 1/8. We show that every graph of size m ≥ 1 has a bipartition in which the Edwards bound holds, and in addition each vertex class contains at most m/4 + √ m/32 + 1/256 − 1/16 edges. This is exact for complete graphs of odd order, which we show are the only extremal graphs without isolated...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1993