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Draw K4,5 and C6; chromatic numbers

Short answer

K4,5 is a complete bipartite graph with partitions of sizes 4 and 5. Its chromatic number is 2. C6 is an even cycle of length 6, which is bipartite, so its chromatic number is 2.

Draw the figure for the complete bipartite graph K4,5 and the cycle graph with 6 vertices (This is usually denoted by C6). What is the chromatic number of the drawn complete bipartite graph K4,5 and the cycle graph C6.

Complete bipartite graph K₄,₅

K₄,₅ (complete bipartite graph): A = {a1, a2, a3, a4}, B = {b1, b2, b3, b4, b5}; each of the 4 vertices of A is joined to each of the 5 vertices of B, giving 20 edges.

   a1      a2      a3      a4         set A
   | \ \ \ / | \ \ \ ... (every a joined to every b)
   b1    b2    b3    b4    b5         set B
Edge present?b1b2b3b4b5
a1..a4yesyesyesyesyes

Coloring: color 1 for all of A, color 2 for all of B. The graph has edges, so one color is not enough. χ(K₄,₅) = 2.

Cycle C₆

C₆ (cycle with 6 vertices): v1..v6 with edges v1v2, v2v3, v3v4, v4v5, v5v6, v6v1.

        v1
       /  \
     v6    v2
      |    |
     v5    v3
       \  /
        v4

Coloring: alternate two colors around the cycle.

Vertexv1v2v3v4v5v6
Color121212

v6 (color 2) is adjacent to v5 (1) and v1 (1), so it is proper. One color is not enough. χ(C₆) = 2 (even cycle).

Summary: χ(K₄,₅) = 2 and χ(C₆) = 2.

✓ Answer checked by Claude Opus 5.5 (high effort)

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