NDLI: On judicious partitions of graphs

Content Provider	Springer Nature Link
Author	Xu, Baogang Liu, Muhuo
Copyright Year	2015
Abstract	Let $$k, m$$ be positive integers, let $$G$$ be a graph with $$m$$ edges, and let $$h(m)=\sqrt{2m+\frac{1}{4}}-\frac{1}{2}$$ . Bollobás and Scott asked whether $$G$$ admits a $$k$$ -partition $$V_{1}, V_{2}, \ldots , V_{k}$$ such that $$\max _{1\le i\le k} \{e(V_{i})\}\le \frac{m}{k^2}+\frac{k-1}{2k^2}h(m)$$ and $$e(V_1, \ldots , V_k)\ge {k-1\over k} m +{k-1\over 2k}h(m) -\frac{(k-2)^{2}}{8k}$$ . In this paper, we present a positive answer to this problem on the graphs with large number of edges and small number of vertices with degrees being multiples of $$k$$ . Particularly, if $$d$$ is not a multiple of $$k$$ and $$G$$ is $$d$$ -regular with $$m\ge {9\over 128}k^4(k-2)^2$$ , then $$G$$ admits a $$k$$ -partition as desired. We also improve an earlier result by showing that $$G$$ admits a partition $$V_{1}, V_{2}, \ldots , V_{k}$$ such that $$e(V_{1},V_{2},\ldots ,V_{k})\ge \frac{k-1}{k}m+\frac{k-1}{2k}h(m)-\frac{(k-2)^{2}}{2(k-1)}$$ and $$\max _{1\le i\le k}\{e(V_{i})\}\le \frac{m}{k^{2}}+\frac{k-1}{2k^{2}}h(m)$$ .
Ending Page	1398
Page Count	16
Starting Page	1383
File Format	PDF
ISSN	13826905
e-ISSN	15732886
Journal	Journal of Combinatorial Optimization
Issue Number	4
Volume Number	31
Language	English
Publisher	Springer US
Publisher Date	2015-01-20
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Partition Graph Judicious Convex and Discrete Geometry Structural characterization of families of graphs Extremal problems Theory of Computation Mathematical Modeling and Industrial Mathematics Operation Research/Decision Theory Combinatorics Optimization
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Control and Optimization Discrete Mathematics and Combinatorics Computational Theory and Mathematics Computer Science Applications

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