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Mathematics Homework Solutions
#28051
Let G, H be graphs such that G is a subgraph of H. Prove or disprove each of the following: (a) alpha(G) <= alpha(H) (b) alpha(G) >= alpha(H) (c) omega(G) <= omega(H) (d) omega(G) >= omega(H)
The stability number, alpha(G), of a graph G is the cardinality of the largest subset S of V(G), the vertex set of G, such that no two of the vertices in S are connected by an edge of G.
The clique number, omega(G), of a graph G is the cardinality of the largest subset S of V(G), the vertex set of G, such that every pair of vertices in S are connected by an edge of G.
Let G, H be graphs such that G is a subgraph of H. Prove or disprove each of the following:
(a) alpha(G) <= alpha(H)
(b) alpha(G) >= alpha(H)
(c) omega(G) <= omega(H)
(d) omega(G) >= omega(H)
Step-by-step proofs are given for the given statements that are true. Counterexamples (with detailed justifications) for the given statements that are false are provided in an attached .doc file.
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