Article

# Three ways to cover a graph

Discrete Mathematics 05/2012; 339(2). DOI: 10.1016/j.disc.2015.10.023

Source: arXiv

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**ABSTRACT:**A new, constructive proof with a small explicit constant is given to the Erd\H{o}s-Pyber theorem which says that the edges of a graph on $n$ vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at most $O(n/\log n)$ times. The theorem is generalized to uniform hypergraphs. Similar bounds with smaller constant value is provided for fractional partitioning both for graphs and for uniform hypergraphs. We show that these latter constants cannot be improved by more than a factor of 1.89 even for fractional covering by arbitrary complete multipartite subgraphs or subhypergraphs. In the case every vertex of the graph is connected to at least $n-m$ other vertices, we prove the existence of a fractional covering of the edges by complete bipartite graphs such that every vertex is covered at most $O(m/\log m)$ times, with only a slightly worse explicit constant. This result also generalizes to uniform hypergraphs. Our results give new improved bounds on the complexity of graph and uniform hypergraph based secret sharing schemes, and show the limits of the method at the same time.Graphs and Combinatorics 11/2013; 31(5). DOI:10.1007/s00373-014-1448-7 · 0.39 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We investigate edge-intersection graphs of paths in the plane grid, regarding a parameter called the bend-number. I.e., every vertex is represented by a grid path and two vertices are adjacent if and only if the two grid paths share at least one grid-edge. The bend-number is the minimum $k$ such that grid-paths with at most $k$ bends each suffice to represent a given graph. This parameter is related to the interval-number and the track-number of a graph. We show that for every $k$ there is a graph with bend-number $k$. Moreover we provide new upper and lower bounds of the bend-number of graphs in terms of degeneracy, treewidth, edge clique covers and the maximum degree. Furthermore we give bounds on the bend-number of $K_{m,n}$ and determine it exactly for some pairs of $m$ and $n$. Finally, we prove that recognizing single-bend graphs is NP-complete, providing the first such result in this field.Discrete Applied Mathematics 04/2014; 167:144-162. DOI:10.1016/j.dam.2013.10.035 · 0.80 Impact Factor