query: Prove that every planar graph is the union of at most $5$ acyclic graphs.
positive_passages.docid: 27407
positive_passages.text: every planar graph can be expressed as a union of five edge-disjoint forests
negative_passages.docid: 89164
negative_passages.text: When is the graph of the divisor relation on $[n]$ planar?
negative_passages.docid: 60492
negative_passages.text: Why girth of Petersen Graph is five? (Proof)
negative_passages.docid: 6429
negative_passages.text: prove that a connected graph with $n$ vertices has at least $n-1$ edges
negative_passages.docid: 41726
negative_passages.text: Musical and combinatorial proof
negative_passages.docid: 78243
negative_passages.text: graph-theory problem about outdegree and indegree
negative_passages.docid: 100978
negative_passages.text: If a graph with $n$ vertices and $n$ edges there must a cycle?
negative_passages.docid: 76029
negative_passages.text: Induction proof: n lines in a plane
negative_passages.docid: 122998
negative_passages.text: number of edges induction proof
negative_passages.docid: 48184
negative_passages.text: Prove a graph Containing $2k$ odd vertices contains $k$ distinct trails
negative_passages.docid: 121402
negative_passages.text: Planar Realization of a Graph in Three-Space
negative_passages.docid: 113015
negative_passages.text: Eulerian Paths in a Complete Graph $K_5$
negative_passages.docid: 94838
negative_passages.text: Show that a self-complementary graph contains an odd number of vertices with degree $(n-1)/2$
negative_passages.docid: 100466
negative_passages.text: The number of labelled graphs with all vertices of even degree
negative_passages.docid: 145181
negative_passages.text: How can I prove the maximum number of edges?
negative_passages.docid: 3439
negative_passages.text: Other proofs that subgroups of $A_5$ have order at most 12
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 $1$