An Improved Bound for Vertex Partitions by Connected Monochromatic K-Regular Graphs
نویسندگان
چکیده
Improving a result of Sárközy and Selkow, we show that for all integers r, k ≥ 2 there exists a constant n0 = n0(r, k ) such that if n ≥ n0 and the edges of the complete graph Kn are colored with r colors then the vertex set of Kn can be partitioned into at most 100r log r + 2rk vertex disjoint connected monochromatic k-regular subgraphs and vertices. This is close to best possible. C © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 127–145, 2013 Contract grant sponsor: NSF; Contract grant Number: DMS-0968699. Journal of Graph Theory C © 2012 Wiley Periodicals, Inc. 127 128 JOURNAL OF GRAPH THEORY
منابع مشابه
k-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 کاملVertex Partitions by Connected Monochromatic k-Regular Graphs
Generalizing a result of Erdo s, Gya rfa s and Pyber we show that there exists a constant c such that for any integers r, k 2 and for any coloring of the edges of a complete graph with r colors, its vertices can be partitioned into at most r log r+k) connected monochromatic k-regular subgraphs and vertices. We also show that the same result holds for complete bipartite graphs, generalizing a re...
متن کاملLarge Monochromatic Components in Edge Colored Graphs with a Minimum Degree Condition
It is well-known that in every k-coloring of the edges of the complete graph Kn there is a monochromatic connected component of order at least n k−1 . In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For k = 2 the authors proved that δ(G) > 3n 4 ensures a monochromatic connected component with at least δ(G) + 1 vertices in every...
متن کاملON GENERALIZED k-DIAMETER OF k-REGULAR k-CONNECTED GRAPHS
In this paper, motivated by the study of the wide diameter and the Rabin number of graphs, we define the generalized k-diameter of k-connected graphs, and show that every k-regular k-connected graph on n vertices has the generalized k-diameter at most n/2 and this upper bound cannot be improved when n = 4k − 6 + i(2k − 4).
متن کاملOn Alliance Partitions and Bisection Width for Planar Graphs
An alliance in a graph is a set of vertices (allies) such that each vertex in the alliance has at least as many allies (counting the vertex itself) as non-allies in its neighborhood of the graph. We show how to construct infinitely many non-trivial examples of graphs that can not be partitioned into alliances and we show that any planar graph with minimum degree at least 4 can be split into two...
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:
- Journal of Graph Theory
دوره 73 شماره
صفحات -
تاریخ انتشار 2013