Question 8 (20 marks) denote by x(g) the chromatic mumber of a graph g (a) let g be a graph with no 3-cycle. let ei and e2 be two distinct edges of g with a common end-vertex. let h = g - (e1,e2} be the spanning subgraph of g obtained from g by deleting e and e2. prove that x(g)-1x(h) x(g). (b) suppose that g is a graph and v is a cut-vertex of g. prove that there exist subgraphs gi and g2 of g such that g giug2, v(g)n v(g2) = {v} and max{x(gi), x(g2)}. x(g)

the central idea is that:

individuals should refuse to cooperate with institutions they believe unjust.

thoreau felt  that governments were more often agents of corruption than justice, even democracies, and that just because the majority feels a certain way does not make it just or correct in absolute terms. he uses the fact that slavery exists in our democracy to support this idea.

he goes on to say that, as a result, people should live true to their own internal laws regarding right and wrong, as a product of reflection and a moral life, rather than following every law that society imposes. 

step-by-step explanation:

7

step-by-step explanation:

rx and sx

step-by-step explanation: