query: Let $G$ be a graph. Let $k$ be the minimal degree of $G$. Show that $G$ contains a cycle of length $k + 1$.
positive_passages.docid: 101520
positive_passages.text: Let $G$ be a graph of minimum degree $k>1$. Show that $G$ has a cycle of length at least $k+1$
negative_passages.docid: 91134
negative_passages.text: In any finite graph with at least two vertices, there must be two vertices with the same degree
negative_passages.docid: 29467
negative_passages.text: Prove that the minimum number of cycles is $m-n+1$
negative_passages.docid: 94406
negative_passages.text: Degree of minimum polynomial at most $n$ without Cayley-Hamilton?
negative_passages.docid: 129752
negative_passages.text: How to find whether it is possible for each vertex of a graph to have a different degree?
negative_passages.docid: 17268
negative_passages.text: Cycles in permutations of a given size
negative_passages.docid: 127965
negative_passages.text: Graph theory (Graph Connectivity)
negative_passages.docid: 64269
negative_passages.text: Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$
negative_passages.docid: 145718
negative_passages.text: Proving that A connected graph on $n$ vertices is a tree iff it has $n-1$ edges
negative_passages.docid: 131137
negative_passages.text: If ${\lvert{E(G)}\rvert}>{\lvert{V(G)}\rvert}+3$, graph $G$ has at least two disjoint cycles.
negative_passages.docid: 120558
negative_passages.text: If $H$ has maximum vertex degree at most $3$ and is a minor of $G$, then $H$ is a topological minor of $G$.
negative_passages.docid: 83036
negative_passages.text: f(g(x)) has a degree divisible by n
negative_passages.docid: 82295
negative_passages.text: Showing that the minimal polynomial of an $n \times n$ matrix has degree at most $n$ without using the Cayley-Hamilton Theorem
negative_passages.docid: 119808
negative_passages.text: Proving that $n$-cycle has order $n$
negative_passages.docid: 107716
negative_passages.text: Show that $G$ is cyclic
negative_passages.docid: 140199
negative_passages.text: Proof that disjoint cycles commute
negative_passages.docid: 28877
negative_passages.text: Graph with $k$ components.