query: Does every 2-coloring of the edges of K6 contain 2 monochromatic triangles?
positive_passages.docid: 1000
positive_passages.text: $K_6$ contains at least two monochromatic $K_3$ graphs.
negative_passages.docid: 122998
negative_passages.text: number of edges induction proof
negative_passages.docid: 35458
negative_passages.text: Prove, that graph $G$ has at least $\chi(G)(\chi(G)-1)/2$ edges.
negative_passages.docid: 28877
negative_passages.text: Graph with $k$ components.
negative_passages.docid: 39753
negative_passages.text: Proving that in a simple graph with $n$ vertices, two vertices have the same degree
negative_passages.docid: 146933
negative_passages.text: Proving bipartition in a connected planar graph
negative_passages.docid: 141482
negative_passages.text: Proof that plane with $N$ lines can be painted with two colors so that any two neighboring regions are painted in different colors
negative_passages.docid: 123567
negative_passages.text: Number of permutations which are products of exactly two disjoint cycles.
negative_passages.docid: 94928
negative_passages.text: Every $k$ vertices in an $k$ - connected graph are contained in a cycle.
negative_passages.docid: 78110
negative_passages.text: A connected k-regular bipartite graph is 2-connected.
negative_passages.docid: 3479
negative_passages.text: How to count the no. of distinct ways in which $1,2,...,6$ can be assigned to $6$ faces of a cube?
negative_passages.docid: 131031
negative_passages.text: Q: Proving G is d-colorable
negative_passages.docid: 124467
negative_passages.text: Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k[X_1,X_2,X_3,X_4]$?
negative_passages.docid: 129935
negative_passages.text: Coloring problem with equilateral triangles
negative_passages.docid: 90747
negative_passages.text: Upper bound on $\chi(G)$ for a triangle-free graph
negative_passages.docid: 23916
negative_passages.text: Does 3-partite graph with at least n+1 edges per vertex have a triangle?
negative_passages.docid: 92267
negative_passages.text: Adjacency matrix and existence of triangle