---
title: "Draw K6 and its chromatic number"
subject: "Discrete Structure"
chapter: 4
key: "semester_4/discrete_structure:draw-k6-chromatic-number"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/draw-k6-chromatic-number
site: BCT Engineering
---

# Draw K6 and its chromatic number

Discrete Structure · Chapter 4: Graph Theory

- Asked in: 2071 Bhadra, 2+2 marks
- Video: [29 - Graph Coloring and Chromatic Number of a Graph](https://www.youtube.com/watch?v=G-dkYSoluys) — KnowledgeGATE by Sanchit Sir
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/discrete_structure/draw-k6-chromatic-number
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-4

## Question

Draw the figure for the complete graph with 6 vertices (This is usually denoted by K6). What is the chromatic number of the complete graph K6?

## Short answer

K6 has six vertices all pairwise adjacent, so six different colors are needed. No two vertices can share a color, giving a chromatic number χ(K6) = 6.

## Answer

**K₆ (complete graph on 6 vertices):** 6 vertices, every pair joined, so 15 edges and every vertex has degree 5.

```text
            v1
          /    \
        v6      v2        hexagon v1..v6 plus all 9 diagonals:
        | \    / |        v1v3 v1v4 v1v5 v2v4 v2v5 v2v6
        |  X  X  |        v3v5 v3v6 v4v6
        v5      v3
          \    /
            v4
```
Edges: all pairs vi vj with 1 ≤ i < j ≤ 6.

**Coloring:** each vertex is adjacent to all five others, so six different colors are needed.

| Vertex | v1 | v2 | v3 | v4 | v5 | v6 |
|---|---|---|---|---|---|---|
| Color | 1 | 2 | 3 | 4 | 5 | 6 |

Six colors are enough and fewer cannot work. So **χ(K₆) = 6**. In general χ(Kₙ) = n.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. K6 has how many edges?
   A. 15
   B. 12
   C. 30
   D. 6
   Answer: A. 6 x 5/2 = 15. (Key point: Edges)

2. Each vertex of K6 has degree:
   A. 6
   B. 5
   C. 4
   D. 3
   Answer: B. It is joined to all other 5 vertices. (Key point: Degree)

3. chi(K6) is:
   A. 5
   B. 3
   C. 6
   D. 2
   Answer: C. All vertices are mutually adjacent so six distinct colors are needed. (Key point: Chromatic)

## Close variants set in other exams

- [Draw K5 and its chromatic number](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Adraw-k5-chromatic-number) — Discrete Structure, chapter 4, asked 4×
- [Draw K3,4 and C5; chromatic numbers](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Adraw-k34-c5-chromatic) — Discrete Structure, chapter 4, asked 4×
- [Draw K4,5 and C6; chromatic numbers](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Adraw-k45-c6-chromatic) — Discrete Structure, chapter 4, asked 4×

## Related in Discrete Structure

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