query: number of cycles in the graph is at least $k − n + 1$
positive_passages.docid: 29467
positive_passages.text: Prove that the minimum number of cycles is $m-n+1$
negative_passages.docid: 117685
negative_passages.text: Prove that a bipartite planar graph has a vertex of degree at most 3
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: 49465
negative_passages.text: How many Hamiltonian cycles are there in a complete graph $K_n$ ($n\geq 3$) Why?
negative_passages.docid: 90618
negative_passages.text: Color the edges of $K_6$ red or blue. Prove that there is a cycle of length 4 with monochromatic edges.
negative_passages.docid: 94928
negative_passages.text: Every $k$ vertices in an $k$ - connected graph are contained in a cycle.
negative_passages.docid: 72267
negative_passages.text: Prove there's a simple path of length $k$ in a simple graph $G$ where all the vertices have degree of at least $k$
negative_passages.docid: 117349
negative_passages.text: counting triangles in a graph or its complement
negative_passages.docid: 114752
negative_passages.text: Number of permutations of a specific cycle decomposition
negative_passages.docid: 61340
negative_passages.text: Prove that there exists bipartite graph with this degree sequence: $(3,3,3,3,3,5,6,6,6,6,6,6,6,6)$
negative_passages.docid: 152653
negative_passages.text: Count Eulerian cycles in $K_{n,2}$
negative_passages.docid: 9009
negative_passages.text: $n$ choose $k$ where $n$ is less than $k$
negative_passages.docid: 155372
negative_passages.text: Find the number of permutations in $S_n$ containing fixed elements in one cycle
negative_passages.docid: 1000
negative_passages.text: $K_6$ contains at least two monochromatic $K_3$ graphs.
negative_passages.docid: 35458
negative_passages.text: Prove, that graph $G$ has at least $\chi(G)(\chi(G)-1)/2$ edges.
negative_passages.docid: 59247
negative_passages.text: How many connected graphs over V vertices and E edges?
negative_passages.docid: 25382
negative_passages.text: Maximum number of edges in a bipartite graph