Graph Theory - 06/03/2026

A bipartite graph can also be interpreted as a \(2\)-colorable graph. That means every vertex is assigned one of two possible colors, such that no adjacent vertices share the same color. This can be generalized to any number of colors, so a graph G can be k-colorable if you assign a color \(c \in \{c1,c2,\dots,c_{k}\}\) to every vertex such that no adjacent vertices share the same color. Figures 1 and 2 show an example of a \(2\)-colorable and \(3\)-colorable graph, respectively.

Figure 1
\(2\)-colorable graph
Figure 2
\(3\)-colorable graph

Which of the following statements are true

I. Cycles with odd number of vertices cannot be \(2\)-colorable;

II. A complete graph with \(N\) vertices is \(N\)-colorable at minimum;

III. An empty graph is not \(2\)-colorable;

IV. A graph is \( (k+1) \)-colorable if and only if it is \(k\)-colorable.

A) I, II

B) I, II, III

C) I, II, III, IV

D) III, IV

E) None of the above

Original idea by: Fernando de Facio Rossetti

Comentários

  1. Joao Meidanis7 de março de 2026 às 12:19

    What's an empty graph?

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    Respostas
    1. Fernando7 de março de 2026 às 13:03

      "An empty graph is a graph in which no two vertices are adjacent, that is, one whose edge set is empty" (Bondy & Murty, Graph Theory, page 4)

      Excluir

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