Question

5.1.39. (!) Prove that every $k$-chromatic graph has at least $\binom{k}{2}$ edges. Use this to prove that if $G$ is the union of $m$ complete graphs of order $m$, then $\chi(G) \le 1 + m\sqrt{m} - 1$. Can you explain proof with sketching a graph with the proof.

          5.1.39. (!) Prove that every $k$-chromatic graph has at least $\binom{k}{2}$ edges. Use this to prove that if $G$ is the union of $m$ complete graphs of order $m$, then $\chi(G) \le 1 + m\sqrt{m} - 1$.
Can you explain proof with sketching a graph with the proof.

5.1.39. (!) Prove that every k-chromatic graph has at least k2 edges. Use this to prove that if G is the union of m complete graphs of order m, then χ(G) ≤ 1 + m√(m) - 1.
Can you explain proof with sketching a graph with the proof.

Added by Robert W.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Suchitra K

07/01/2022

Step 1

Step 1: First, we prove that every k-chromatic graph has at least ([k],[2]) edges. Show more…

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5.1.39. (!) Prove that every k-chromatic graph has at least ([k],[2]) edges. Use this to prove that if G is the union of m complete graphs of order m, then chi (G)<=1+msqrt(m-1). Can you explain proof with sketching a graph with the proof. 5.1.39. Prove that every k-chromatic graph has at least () edges. Use this to prove that if G is the union of m complete graphs of order m, then x(G) 1 + m/m - 1 Can you explain proof with sketching a graph with the proof.