query: every planar graph can be expressed as a union of (maximum) five forests
positive_passages.docid: 27407
positive_passages.text: every planar graph can be expressed as a union of five edge-disjoint forests
negative_passages.docid: 101033
negative_passages.text: Algorithm for procedureral 2D map with connected paths
negative_passages.docid: 134242
negative_passages.text: Graph partition that span a third of edges
negative_passages.docid: 158856
negative_passages.text: Prove that a graph with the same number of edges and vertices contains one cycle
negative_passages.docid: 59247
negative_passages.text: How many connected graphs over V vertices and E edges?
negative_passages.docid: 14438
negative_passages.text: Trees with explicit formula
negative_passages.docid: 49879
negative_passages.text: A bipartite graph with a perfect matching has a vertex with each edge contained in a perfect matching
negative_passages.docid: 48692
negative_passages.text: A problem about the graph theory
negative_passages.docid: 45510
negative_passages.text: Number of vertices present in all maximum matchings
negative_passages.docid: 72417
negative_passages.text: Bipartite graphs
negative_passages.docid: 119304
negative_passages.text: Latin Square and Graph Theory
negative_passages.docid: 59780
negative_passages.text: how to show that when an edge is removed from K5, the resluting subgraph is planar.
negative_passages.docid: 123103
negative_passages.text: Prove recursive formula for number of spanning trees in complete graph.
negative_passages.docid: 123626
negative_passages.text: Prove that even $n$ can be partitioned to $\frac n2$ edges
negative_passages.docid: 110422
negative_passages.text: Prove that a graph $G$ is a forest if and only if every induced subgraph of $G$ contain a vertex of degree at most $1$
negative_passages.docid: 28877
negative_passages.text: Graph with $k$ components.
negative_passages.docid: 116538
negative_passages.text: Proof of Connected Diagrams