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Theorem 2.1 For every graph G, o (G) ≤ x (G). Proof It suffices to assume that G is a nontrivial connected graph. Suppose that x (G) = k and A(G) = A, and let d = A + 1. Let c' be a proper k-coloring of G, using the colors 1, 2, ..., k. So adjacent vertices of G are colored differently by c'. Define a k-coloring c of G by c(v) = di-1 if c'(v) = i. We show that c is a sigma coloring of G, which implies that o (G) ≤ k = x (G). Let u and u be two adjacent vertices of G, where c'(u) =s and c'(v) = t. Then st. Let S be the multiset of colors assigned by c' to the neighbors of u and let T be the multiset of colors assigned by c' to the neighbors of v. Since se T-S and t€ S-T, it follows that S T. Hence there is a largest integer j with max{s, t} ≤ j≤ k such that S and T contain j an unequal number of times. Suppose that S contains j a total of a times and T contains j a total of b times, where say 0≤a

Theorem 2.1 For every graph G, o (G) ≤ x (G). Proof It suffices to assume that G is a nontrivial connected graph. Suppose that x (G) = k and A(G) = A, and let d = A + 1. Let c' be a proper k-coloring of G, using the colors 1, 2, ..., k. So adjacent vertices of G are colored differently by c'. Define a k-coloring c of G by c(v) = di-1 if c'(v) = i. We show that c is a sigma coloring of G, which implies that o (G) ≤ k = x (G). Let u and u be two adjacent vertices of G, where c'(u) =s and c'(v) = t. Then st. Let S be the multiset of colors assigned by c' to the neighbors of u and let T be the multiset of colors assigned by c' to the neighbors of v. Since se T-S and t€ S-T, it follows that S T. Hence there is a largest integer j with max{s, t} ≤ j≤ k such that S and T contain j an unequal number of times. Suppose that S contains j a total of a times and T contains j a total of b times, where say 0≤a

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Theorem 2.1 For every graph G, o (G) ≤ x (G). Proof It suffices to assume that G is a nontrivial connected graph. Suppose that x (G) = k and A(G) = A, and let d = A + 1. Let c' be a proper k-coloring of G, using the colors 1, 2, ..., k. So adjacent vertices of G are colored differently by c'. Define a k-coloring c of G by c(v) = di- if c'(v) = i. We show that c is a sigma coloring of G, which implies that o (G) ≤ k = x (G). Let u and u be two adjacent vertices of G, where c'(u) =s and c'(v) = t. Then st. Let S be the multiset of colors assigned by c' to the neighbors of u and let T be the multiset of colors assigned by c' to the neighbors of v. Since se T-S and t€ S-T, it follows that S T. Hence there is a largest integer j with max(s, t) ≤ j≤ k such that S and T contain j an unequal number of times. Suppose that S contains j a total of a times and T contains j a total of b times, where say 0 ≤a <b≤A. We claim that o (u) o (v) for the coloring c. Let r= while k-j-1 Σadk-1-i, i=0 where ao, a1,..., ak-j-1 are nonnegative integers. From the defining property of j, it follows that o(u) = r + adj-¹ + p₁ (d) and o(v) = r + bd³-¹ + p2(d), where pi(d) and p2 (d) are polynomials in d having degree at most j - 2. Thus o(u) = r + adj-¹ + p₁(d) ≤r + adj-¹ + Adj-2 <r+adi-¹ +di-¹ = r + (a + 1)d¹-¹ ≤r+bdj-¹, (2) o (v) = r + bdj−¹ + p2(d) ≥ r + bdj−¹. Thus o (u) < o (v) and so c is a sigma coloring of G.
Transcribed Image Text:Theorem 2.1 For every graph G, o (G) ≤ x (G). Proof It suffices to assume that G is a nontrivial connected graph. Suppose that x (G) = k and A(G) = A, and let d = A + 1. Let c' be a proper k-coloring of G, using the colors 1, 2, ..., k. So adjacent vertices of G are colored differently by c'. Define a k-coloring c of G by c(v) = di- if c'(v) = i. We show that c is a sigma coloring of G, which implies that o (G) ≤ k = x (G). Let u and u be two adjacent vertices of G, where c'(u) =s and c'(v) = t. Then st. Let S be the multiset of colors assigned by c' to the neighbors of u and let T be the multiset of colors assigned by c' to the neighbors of v. Since se T-S and t€ S-T, it follows that S T. Hence there is a largest integer j with max(s, t) ≤ j≤ k such that S and T contain j an unequal number of times. Suppose that S contains j a total of a times and T contains j a total of b times, where say 0 ≤a <b≤A. We claim that o (u) o (v) for the coloring c. Let r= while k-j-1 Σadk-1-i, i=0 where ao, a1,..., ak-j-1 are nonnegative integers. From the defining property of j, it follows that o(u) = r + adj-¹ + p₁ (d) and o(v) = r + bd³-¹ + p2(d), where pi(d) and p2 (d) are polynomials in d having degree at most j - 2. Thus o(u) = r + adj-¹ + p₁(d) ≤r + adj-¹ + Adj-2 <r+adi-¹ +di-¹ = r + (a + 1)d¹-¹ ≤r+bdj-¹, (2) o (v) = r + bdj−¹ + p2(d) ≥ r + bdj−¹. Thus o (u) < o (v) and so c is a sigma coloring of G.
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