query: Proving that a planar graph is bipartite
positive_passages.docid: 146933
positive_passages.text: Proving bipartition in a connected planar graph
negative_passages.docid: 43555
negative_passages.text: Show that a finite regular bipartite graph has a perfect matching
negative_passages.docid: 28877
negative_passages.text: Graph with $k$ components.
negative_passages.docid: 4112
negative_passages.text: How to prove that a simple graph having 11 or more vertices or its complement is not planar?
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: 65639
negative_passages.text: Existence of a self-complementary graph
negative_passages.docid: 50886
negative_passages.text: Two disjoint spanning trees, spanning subgraph with all even degrees
negative_passages.docid: 89452
negative_passages.text: Proof about Graph with no triangle
negative_passages.docid: 123626
negative_passages.text: Prove that even $n$ can be partitioned to $\frac n2$ edges
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: 100289
negative_passages.text: Question about self-complementary graphs
negative_passages.docid: 52261
negative_passages.text: Proof that any self-complementary graph has to have $4k$ or $4k+1$ vertex, for some $k \in \mathbb N$
negative_passages.docid: 131736
negative_passages.text: Proving a triangle with different edge colors exists in a graph.
negative_passages.docid: 78243
negative_passages.text: graph-theory problem about outdegree and indegree
negative_passages.docid: 78110
negative_passages.text: A connected k-regular bipartite graph is 2-connected.
negative_passages.docid: 153550
negative_passages.text: Is there any connection between distributive lattices and planar graphs?
negative_passages.docid: 154564
negative_passages.text: Prove that the sum of minimum edge cover and maximum matching is the vertex count