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| {0}, {1}, {2}, {3}, {4} | {0}, {1}, {2}, {3}, {4} | | {0}, {1}, {2}, {5} | {0}, {1}, {2}, {5} | | {0}, {1}, {6} | {0}, {1}, {6} | | {0}, {7} | {0}, {7} | Table 8.13: The non-essential sets for the graph of Figure 8.12. In order to compute the non-essential sets for this graph we first construct the graph of Figure 8.14 where the vertex set is the set of all subsets of $V(G)$ and two vertices are adjacent if and only if their intersection is empty. It can be seen that the independent sets of this graph are exactly the sets of non-adjacent vertices of *G*. From this graph we can now construct the graph of Figure 8.15 by joining two vertices if and only if their union is not an independent set. The independent sets of this graph are now exactly the essential sets of *G* (by Lemma 8.6). We then delete the vertices corresponding to the essential sets from this graph and find the independent sets of the resulting graph (Figure 8.16). These independent sets correspond to the non-essential sets for *G*. In fact, as an easy exercise we can show that there is a one-to-one correspondence between the non-essential sets for *G* and the independent sets of this final graph.  Figure 8.14: The independent set graph for Figure 8.12.  Figure 8.15: The essential set graph for Figure 8.12.  Figure 8.16: The non-essential set graph for Figure 8.12. It should be noted that it is possible to find a smaller representation for this last graph than that shown above but still have it be useful as an aid to finding the non-essential sets. ## Exercises * Show that if G is an induced subgraph of H then $alpha(G) leq alpha(H)$ . * Show that $alpha(K_{m,n}) = min{ m,n}$ . * Let $K_{m,n}$ be a complete bipartite graph with parts X and Y where $|X| = m$ and $|Y| = n$ . Show that $I(K_{m,n}) = binom{m + n}{m}$ . * Let G be a simple connected graph on at least three vertices such that every vertex has degree at least two. Show that $alpha(G) geq lfloorfrac{n}{2}rfloor$ , where n is the number of vertices. * Let G be a simple connected graph on at least three vertices such that every vertex has degree at least two. Show that $omega(G) geq lceilfrac{n}{2}rceil$ , where n is the number of vertices. * Let G be a simple connected graph on at least three vertices such that every vertex has degree at least two. Show that $chi(G) geq lceilfrac{n}{lfloorfrac{n}{2}rfloor}rceil$ , where n is the number of vertices. * Find all graphs G such that $omega(G) = alpha(G)$ . * Find all graphs G such that $omega(G) = chi(G)$ . * Find all graphs G such that $alpha(G) = chi(G)$ . * Find all graphs G such that $alpha(G) + omega(G) = n$ , where n is the number of vertices. * Find all graphs G such that $alpha(G) + chi(G) = n$ , where n is the number of vertices. * Find all graphs G such that $omega(G) + chi(G) = n$ , where n is the number of vertices. * Let G be a simple connected planar graph on at least three vertices such that every vertex has degree at least two. Show that $omega(G) + chi(G) - alpha(G) = n$ , where n is the number of vertices. * Let G be a simple connected planar graph on at least three vertices such that every vertex has degree at least two. Show that $omega(G) + chi(G) - alpha(G) = m + 1$ , where m is the number of edges. * Let G be any simple connected planar graph on at least three vertices such that every vertex has degree at least two and let k be an integer with $k > 1$ . Show that if every cycle in G has length at least k then χ (G) ≤ k − 1. * Prove Lemma 8.6. * Prove Lemma 8.7. * Prove Corollary 8.10. * Prove Corollary 8.11. * Prove Corollary 8.12. * Prove Corollary 8.13. * Prove Corollary 8.14. * Let $A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z$ be pairwise disjoint subsets of $V(Gamma)$ . Suppose α(Γ) = ω(Γ). Find an expression for I(Γ). * Find an expression for I(Kn ). * Find an expression for I(Cn ). * Find an expression for I(Pn ). * Find an expression for I(Cn ) − I(Pn ). * Find an expression for I(Cn ) + I(Pn ). * Find an expression for I(Cn )I(Pn ). * Prove Lemma 8.19. ## Notes The notation *I*(Γ), introduced in Section 8.9, was first used by van Lint and Wilson [121]. They proved some general results concerning *I*(Γ), including Lemmas 8.18 and 8.19, which are also found in [121]. In [121] they also give formulas for many specific types of graphs including *K_{m,n}* , *C_{n}* , *P_{n}* , *Q_{k}* (the k-dimensional hypercube), etc., which are listed below. | Graph | Expression for *I*(Γ) | | $K_{m,n}$ | ${(m + n)}^{m}$ | | $C_{2k}$ | ${(k - 1)}^{k}{(k + 1)}^{k}$ | | $C_{2k + 1}$ | ${(k - 1)}^{k}{(k + 2)}^{k + 1}$ | | *P_{n}* | ${(n - k)}^{k}{(n - k + 1)}^{k + 1}cdots(n - 1)^{n - k - 1}(n)^{n - k}$ | | $Q_{k}$ | ${(frac{3}{2})}^{k2^{k - 1}}{(2)}^{k2^{k - 1}}$ | The problem presented by Problem Set Exercise (18), finding an expression for *I*(Γ), was posed by van Lint and Wilson [121] and has been extensively studied since then; see [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52] for further results concerning this problem. The proof given here for Lemma (11), which shows how to construct larger graphs from smaller ones so as to preserve certain properties (such as independence number or clique number), was first given by Kostochka and Yancey [75]. This lemma plays a key role in their paper proving results about Turán-type problems concerning independence number and clique number. ## Solutions ### Section Exercises #### Section Exercise (11) First we show that every subset containing four or more elements must contain a pair whose sum is even. Suppose we have five integers from our list: *x*, *y*, *z*, *u*, and *v*. At most two can have odd parity (odd or even). If three are even then obviously there will be a pair whose sum is even (since even plus even is even). If two are odd then there will again be a pair whose sum is even (since odd plus odd is even). Now we consider our list: > −9 −7 −5 −3 −1 > > −9 −7 −5 −3 > > −9 −7 −5 −1 > > −9 −7 −3 −1 > > −9 −5 −3 −1 > > −7 −5 −3 −1 > > If we choose any four integers from our list then we will have one of these possibilities listed above; therefore our proof holds. #### Section Exercise (17) We note first from Exercise (16) above that $mathit{omega}(G)mathit{alpha}(G)mathit{chi}(G),,text{=},,mathit{n}^{2}$ . Since χ(*G*) ≥ ω(*G*) we know immediately from this equation that χ(*G*) ≥ √(*n*) ≥ √(*α*) (*G*) ≥ α(*G*) ≥ ω(*G*) ≥ √(*α*) (*G*) ≥ √(*α*) (*G*) ≥ ω(*G*) which implies χ(*G*) ≥ α(*G*) ≥ ω(*G*) as desired. #### Section Exercise (19) Since