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(1) HOMEWORK Let G be a simple graph where the vertices correspond to each of the squares of an 8 8 chess board and where two squares are adjacent if, and only if, a knight can go from one square to the other in one move. What is/are the possible degree(s) of a vertex in G? How many vertices have each degree? How many edges does G have? Solution: A knight moves two squares horizontally and one square vertically, or two squares vertically and one square horizontally, on a chessboard. By examining all possibilities, we see that; the four corner vertices of the graph have degree 2; the eight edge vertices that are next to the corners have degree 3; twenty vertices, the remaining 16 edge vertices plus four more that are next to the corners but not on the edge, have degree 4; sixteen vertices have degree 6; the remaining 16 interior vertices have degree 8. The preceding paragraph answers the rst two questions. For the last question we appeal to the Hand-Shaking Theorem. 1 1 dG (v ) = (4 2 + 8 3 + 20 4 + 16 6 + 16 8) = 168 |E (G)| = 2 2
v V (G)
(2) Let G be a graph with n vertices and exactly n 1 edges. Prove that G has either a vertex of degree 1 or an isolated vertex. Solution: Another way to state this problem would be: Prove that G has at least one vertex of degree less than 2. We will prove this by contradiction, that is we will start out by assuming that the degree of each of the vertices of G is greater than or equal to 2. Under this assumption, the Hand-Shaking Theorem gives us 2n 2 = 2(n 1) = 2|E (G)| =
v V (G)
dG (v ) 2 + 2 + + 2 = 2n
n times
But this is clearly impossible. Therefore our assumption must be false and there must be at least one vertex that has degree less than 2. (3) Prove that if a graph G has exactly two vertices u and v of odd degree, then G has a u, v -path. Solution: We will prove the statement by contradiction. That is, we rst assume that G is a graph with exactly two vertices of odd degree, u and v , and that there is no u, v -path in G.
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By denition, G is a disconnected graph and the vertices u and v must lie in separate connected components of G. Thus the vertex u lies in H , a connected subgraph of G, and all other vertices in H (which, of course, are also vertices in G) have even degree. Therefore, the sum of the degrees of the vertices in H is an odd number. But by the Hand-Shaking Theorem, the sum of the degrees of the vertices in H (as in any graph) must be an even number. So we have a contradiction and our assumption, that there is no u, v -path in G, must be false. (4) Let G be a simple graph. Show that either G or its complement G is connected. Solution: Let G be a simple graph that is not connected and let G be the complement of G. If u and v are vertices in G then they are also vertices in G. If u and v lie in separate connected components of G they are not adjacent in G. Therefore, they must be adjacent in G. Hence there exists a u, v -path in G. If u and v belong to the same connected component of G and w is a vertex in a dierent connected component of G, then both u and v are adjacent w in G. If e1 is the edge of G with endvertices u and w and e2 is the edge of G with endvertices w and v , then (u, e1 , w, e2 , v ) is a u, v -path in G. Since there exists a path between any two vertices of G it is a connected graph. (5) Are any of the graphs Nn , Pn , Cn , Kn and Kn,n complements of each other? Solution: Clearly, Nn , the simple graph with n vertices and no edges, and Kn the simple graph with n vertices and the maximal possible edges, are complements of each other when n > 1. By convention P1 = N1 , so P1 = N1 = N 1 = P 1 . Also, P2 = K2 , thus P2 and N2 are complements of each other. The complement of P3 is not connected and is clearly not the same as any of the graphs on the original list. The graph P4 is isomorphic to its complement (see Problem 6). In general, the graph Pn has n 2 vertices of degree 2 and 2 vertices of degree 1. Therefore P n has n 2 vertices of degree n 3 and 2 vertices of degree n 2. This rules out any matches for P n when n 5. For n = 1 or 2, Cn is not simple. So we dont talk about complements in those cases. Since C3 = K3 , C 3 = N3 . As C 4 is not connected it can not be identied with any of the original graphs. The graph C5 is its own complement (again see Problem 6). We now examine C n when n 6. The graph Cn is 2-regular. Therefore C n is (n 3)-regular. Now, the graph Nn is 0-regular and the graphs Pn and Cn are not regular at all. So no matches so far. The only complete graph with the same number of vertices as C n is n 1-regular. For n even, the graph K n , n does have the same number vertices of as C n , but it 22 is n-regular. Hence, we have no matches for the complement of Cn if n 6.
HOMEWORK 2
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We dont have to examine Kn,n (except perhaps to identify K1,1 with P2 ) since if it was a complement pair with any of the previous graphs on the list we would have already seen that. (6) Show that if a simple graph G is isomorphic to its complement G, then G has either 4k or 4k + 1 vertices for some natural number k . Find all simple graphs on four and ve vertices that are isomorphic to their complements. Solution: If G and G are isomorphic, they must have the same number of edges. However, the total number of edges in G plus the total number of edges in G equals n(n 1) the number of edges in the complete graph on n vertices, which is . Hence, 2 n(n 1) |E (G)| = . 4 This is only possible if n or n 1 is divisible by 4. They cant both be divisible by 4 since one will be odd. So, either n = 4k or n 1 = 4k for some natural number k . In the second case, n = 4k + 1. From the equation above, we see that if a simple graph on four vertices is isomorphic to its complement it must have exactly three edges. This, along with trial and error, shows that the only graph with this property is the path graph P4 . For the ve vertex case, we make heavy use of the graph-theoretic knowledge developed in class so far and quite a bit of logical reasoning. For a simple graph G on ve vertices: (a) The degree of any vertex is either 0, 1, 2, 3 or 4. (b) Since the number of edges must be 54 = 5, the sum of all vertex degrees must 4 be 10. (c) If G G, the vertex degrees (except for 2) appear in pairs. That is, for every = vertex of degree 4 there must be a vertex of degree 0 and for every vertex of degree 3 there must be a vertex of degree 1. This forces the number of vertices of degree 2 to be odd. Also, we can rule out vertices of degree 4 or 0, since in a simple graph on ve vertices if you have a vertex of degree 4 you cannot have a vertex of degree 0. By combining the above observations, we see that set of vertex degrees for a simple graph on ve vertices that is isomorphic to its complement is either {2, 2, 2, 2, 2}, {2, 2, 2, 3, 1} or {2, 3, 1, 3, 1}. With the rst set in mind, we see that C5 , the cycle graph on ve vertices, is isomorphic to its complement. A little more analysis allows us to rule out the second set. If a graph G has exactly one vertex of degree 3 and exactly one vertex of degree 1, then the same will be true for its complement. In fact, it will be the same vertices, with their degrees switched. But in one graph the degree 3 and degree 1 vertices will be adjacent and in the complement they will not. Therefore the graphs cannot be isomorphic.
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This leaves us with the {2, 3, 1, 3, 1} case. With some experimentation, one gets the following example
OOO OOO OOO O // tt // tt t // tt tt // tt t // tt t // tt // ttt tt O OOO
Extra credit if you nd a case Ive missed! (7) The complete bipartite graphs K1,n , known as the star graphs, are trees. Prove that the star graphs are the only complete bipartite graphs which are trees. Solution: Let Km,n be a complete bipartite graph such that m, n > 1. For u1 , u2 , v1 , v2 V (Km,n ), let u1 and u2 be elements of the bipartition set of order m and v1 and v2 be elements of the bipartition set of order n. By denition of the complete bipartite graph, there exists an edge e1 with endvertices u1 and v1 , an edge e2 with endvertices u1 and v2 , an edge e3 with endvertices u2 and v1 and an edge e4 with endvertices u2 and v2 . Therefore c = {u1 , e1 , v1 , e3 , u2 , e4 , v2 , e2 , u1 } is a walk in Km,n with distinct vertices except for the initial and nal vertex, which are the same. That is, c is a cycle in Km,n . Hence, Km,n is not a tree. (8) A graph G is bipartite if there exists nonempty sets X and Y such that V (G) = X Y , X Y = and each edge in G has one endvertex in X and one endvertex in Y . Prove that any tree with at least two vertices is a bipartite graph. Solution: Proof by induction. The only tree on 2 vertices is P2 , which is clearly bipartite. Now assume that every tree on n vertices is a bipartite graph, that is, its vertex set can be decomposed into two sets as described above. Let T be a tree on n + 1 vertices. From class, we know that the vertex set of T contains a leaf, v . Furthermore, T = T v is a tree on n vertices. By the induction hypothesis, the vertex set of T can be decomposed into disjoint sets X and Y . Let w be the neighbor of v in T . Then w is also a vertex in the graph T . WLOG, assume that w X V (T ). Then X = X and Y = Y {v } is a bipartition of V (T ) that makes T a bipartite graph.