JUNIOR INDEPENDENT WORK FALL 2002 ALEXEY SPIRIDONOV This paper represents my own work in accordance with University regulations. 1. Introduction This paper concerns two types of simple graphs: (1) Graphs possessing the maximum possible girth with a given diameter. We derive a complete characterization of these. (2) Graphs, which are both their own duals and their own complements. We present a complete listing for ones drawn in the plane and in the torus. 2. Maximal-girth Graphs 2.1. Definitions. Notation. (u; v)G is the shortest path between u and v with u; v V (G) and 2 (u; v)G E(G). (u; v)G denotes the length of this path. We will omit the subscript⊆ when thek contextk is obvious. Most of the time, we use paths as sets of edges. However, we will occasionally refer to the path's vertex set; by that, we mean the set of vertices that are endpoints of members of the path. The ends (or endpoints) of a path are the two vertices that have degree 1 in its edge set. Definition 2.1. The girth of G, g(G), is the number of edges in the shortest cycle of G. Girth is not defined for forests; therefore, this section excludes them from consideration. Definition 2.2. The diameter of G, d(G) is the maximum of (u; v)G over all u; v V (G). k k 2 Definition 2.3. A d; k { Moore graph is a k-regular graph of diameter d with d 1 i 1 + k − (k 1) vertices. Pi=0 − Definition 2.4. G is a maximal-girth graph if no graph with the same diameter has a larger girth. 2.2. Preliminaries. In this subsection, we present some simple results that will be very useful later. We begin with this trivial lemma: Lemma 2.1. For any G, g(G) 2d(G) + 1. ≤ 1 2 ALEXEY SPIRIDONOV Proof. Suppose there exists G such that g(G) 2d(G) + 2. Let C G be a cycle with V (C) minimum. Let a; b V (C) be two≥ vertices separating⊆ C into two j j 2 paths of length at least d(G) + 1. If (a; b)G d(G), C would not be the shortest k k ≤ cycle of G. Hence, (a; b)G d(G) + 1, which contradicts the definition of the k k ≥ diameter. We thus obtain a more precise characterization of maximal-girth graphs: Corollary 2.2. G is maximal-girth iff g(G) = 2d(G) + 1. Proof. : Suppose g(G) = 2d(G) + 1. By Lemma 2.1, no graph with diameter d(G) and( larger girth can exist. Thus, G is maximal-girth. : Suppose G is maximal-girth, but g(G) = 2d(G)+1. Then, from Lemma 2.1, ) 6 g(G) < 2d(G) + 1. Let G0 be the cycle of length 2d(G) + 1. g(G0) > g(G), but d(G0) = d(G), so G is not maximal-girth. That is a contradiction. Another valuable property of maximal-girth graphs is that: Lemma 2.3. All maximal-girth graphs are 2-vertex-connected. Proof. Suppose G has a cut-vertex v. Then, G v has more than one component. n Assume that some component G0 has the property that v C (G0 v ), where C is the shortest cycle through v. By definition, C has size 2 g(G⊆). Let[u f beg a point g(G) ≥ in C that is at the distance (in C) l = 2 from v. Since C is the shortest cycle through v, u is also l edges away from vin G. Now consider w in a component other g(G) than G0. Since v is a cut-vertex, (u; v) (u; w)G, and d(G) (u; w)G > 2 . Therefore, g(G) < 2d(G) + 1 and G is not⊂ maximal-girth. ≥ k k If no component has a cycle containing v, we may contract all the edges leading out of v to obtain G0. The new graph clearly has the same cycles as G, and v is still a cut-vertex. Therefore, g(G0) = g(G) and d(G0) d(G). We can repeat this procedure indefinitely until v does belong to a cycle in≤ some component, at which point we get the previous case: g(G) = g(G0) < 2d(G0) + 1 2d(G) + 1. ≤ 2.3. Main Result. The following observation motivates our further investigation: Proposition 2.4. All Moore graphs are maximal-girth. Proof. Let G be a d; k { Moore graph, and u V (G). Label Sl the set of points v 2 such that (u; v) = l. Obviously, S0 is 1, and S1 is k. k k j j j j In general, for 0 < i d, every vertex v Si has k edges going to Si 1, Si, or ≤ 2 − Si+1. If v had an edge to Sj with j < i 1, (u; v) would be < i | a contradiction − k k since v Si. If v had an edge to w Sj, with j > i + 1, w would actually be in 2 2 Si+1 | again, a contradiction. Obviously, for each v Si at least one of these k edges must go to a v0 Si 1. 2 2 − If i < d, the others can (obviously) and must go to distinct vertices in Si+1. In other words, (1) Si+1 = (k 1) Si j j − j j Suppose some edge other than (v; v0) goes to Si 1 (0 < i d). Then, the Si i 2 − j ≤ i 1 d j i j ≤ (k 1) Si 1 1, and hence V (G) 1+k j−=0(k 1) +(k(k 1) − 1) j=0− (k − j− j j−d 1 i j j ≤ P − − − P − 1) < 1 + k − (k 1) , so G is not a Moore graph | a contradiction. Similarly, Pi=0 − if we suppose that some edge goes from v to Si (0 < i < d), Si+2 (k 1) Si 2 and G once again cannot be a Moore graph. j j ≤ − j j − JUNIOR INDEPENDENT WORK FALL 2002 3 Moreover, if some v1; v2 Si, 0 < i < d, have a common neighbor w in Si+1, 2 then w has two different neighbors in S(i+1) 1, which was shown to be impossible − above. Therefore, the neighbors of every v1 Si in Si+1 are distinct from those of 2 every other v2 Si. That proves (1). Therefore, every2 cycle through u is of length at least 2d+1 (otherwise, (1) would have to be violated). Since u was chosen arbitrarily, g(G) = 2d + 1. This proposition makes it natural to ask whether there are any non-Moore maximal-girth graphs. Arriving at the answer becomes a relatively simple mat- ter once we describe Moore graphs as follows: Lemma 2.5. A regular maximal-girth graph is a Moore graph. Proof. Suppose G is a k-regular maximal-girth graph. Using the notation from the proof of Proposition 2.4, we want to show that 0 < i < d(G), Si+1 = (k 1) Si . 8 j j − j j As in Proposition 2.4, given v Si with 0 < i d(G), the k edges leaving it go to 2 ≤ Si 1, Si, or Si+1, and at least one, (v; v0), goes to Si 1. − − Suppose some (v; w), v0 = w Si 1 is an edge. Let C0 = (u; v0) (u; w) (w; v) 6 2 − [ [ [ (v0; v). It is the union of two paths, each at most d(G) in length. The paths have the same endpoints, but do not share all vertices. Therefore, C0 contains the edge set of a cycle. The cycle must be at most 2d(G) in length | a contradiction, since G is maximal-girth. Suppose i < d(G) and some (v; w), w Si is an edge. As before, let C0 = 2 (u; v) (u; w) (v; w). C0 then contains the edge set of a cycle, and is of size < 2d(G[) + 1 |[ again, a contradiction. Now we know that k 1 neighbors of every v1 Si are in Si+1; these neighbors − 2 are also distinct from those of every other v2 Si. If that was not the case, the 2 shared neighbor w Si+1 would have two different neighbors in S(i+1) 1, previously shown to be impossible.2 − d(G) d 1 i Therefore, Si+1 = (k 1) Si and V (G) = Si = 1 + k − (k 1) . j j − j j j j Pi=0 j j Pi=0 − Hence, G is a Moore graph. Now we can rephrase our question as whether there are any non-regular maximal- girth graphs. Proposition 2.6. All maximal-girth graphs are regular. Proof. Let G be maximal-girth with d(G) = n, and C | some cycle of size g(G) = 2n + 1 with vertices v0; v1; : : : ; v2n (subscripts in Z=(2n + 1); such a cycle exists by definition of g(G)). Suppose some v = vi has deg v = d + 2 neighbors: (1) (d) (j) vi+1; vi 1; vi ; :::; vi . No vi is in V (C): if it were otherwise, there would exist a cycle shorter− than g(G) (contradicting the definition of girth). (j) Consider u = vi n and some v0 = vi . Define C0 with V (C0) = V (C) v0 − [ f g and E(C0) = E(C) (v; v0) . There are three paths of interest: pC = (u; v0)C , [ f g 0 p ¯ = (E(C) (u; v)) (v; v0), and pG = (u; v0)G.