Spanning Trees with Many Leaves and Average Distance

In this paper we prove several new lower bounds on the maximum number of leaves of a spanning tree of a graph related to its order, independence number, local independence number, and the maximum order of a bipartite subgraph. These new lower bounds were conjectured by the program Graffiti.pc, a variant of the program Graffiti. We use two of these results to give two partial resolutions of conjecture 747 of Graffiti (circa 1992), which states that the average distance of a graph is not more than half the maximum order of an induced bipartite subgraph. If correct, this conjecture would generalize conjecture number 2 of Graffiti, which states that the average distance is not more than the independence number. Conjecture number 2 was first proved by F. Chung. In particular, we show that the average distance is less than half the maximum order of a bipartite subgraph, plus one-half; we also show that if the local independence number is at least five, then the average distance is less than half the maximum order of a bipartite subgraph. In conclusion, we give some open problems related to average distance or the maximum number of leaves of a spanning tree. Introduction and Key Definitions Graffiti, a computer program that makes conjectures, was written by S. Fajtlowicz and dates from the mid-1980’s. Graffiti.pc, a program that makes graph theoretical conjectures utilizing conjecture making strategies similar to those found in Graffiti, was written by E. DeLaViña. The operation of Graffiti.pc and its similarities to Graffiti are described in [10] and [11]; its conjectures can be found in [13]. A numbered, annotated listing of several hundred of Graffiti’s conjectures can be found in [19]. Both Graffiti and Graffiti.pc have correctly conjectured a number of new bounds for several well studied graph invariants; bibliographical information on resulting papers can be found in [12]. the electronic journal of combinatorics 15 (2008), #R33 1 We limit our discussion to graphs that are simple, connected and finite of order n. Although we often identify a graph G with its set of vertices, in cases where we need to be explicit we write V (G). We let α = α(G) denote the independence number of G. If u, v are vertices of G, then σG(u, v) denotes the distance between u and v in G. This is the length of a shortest path in G connecting u and v. The Wiener index or total distance of G, denoted by W = W (G), is the sum of all distances between unordered pairs of distinct vertices of G [16]. Then the average distance of G, denoted by D = D(G), is 2W (G)/[n(n−1)]. Put another way, D(G) is the average distance between pairs of distinct vertices of G. (In the degenerate case n = 1, we set W (G) = D(G) = 0.) Unless stated otherwise, when we refer to a subgraph of a graph G, we mean an induced subgraph. Theorem 1 shown here is the first published result [20] concerning one of the earliest and best known of Graffiti’s conjectures, which states that the average distance of a graph is not more than its independence number. This conjecture is listed as number 2 in [19]. Theorem 1 ([20]). Let G be a graph. Then D < α + 1. Graffiti’s conjecture number 2 was then completely settled by F. Chung in [5], where the following theorem is proved. Theorem 2 ([5]). Let G be a graph. Then D ≤ α, with equality holding if and only if G is complete. In his Ph.D. dissertation [28], the second author generalized Theorem 2 somewhat by characterizing those graphs with order n and independence number α that have maximum average distance, for all possible values of n and α. A different, much shorter proof of this result was later discovered independently by P. Dankelmann [6]. In 1992, Graffiti formulated a new generalization of its own conjecture number 2. This conjecture, stated here as Conjecture 1, is listed as number 747 in [19]. For a graph G, we call the bipartite number of G the maximum order of an (induced) bipartite subgraph. We denote this invariant by b = b(G). (There are many bounds for the maximum number of edges in a bipartite subgraph; for such results, see [1], [3] and [22]. Some results on the bipartite number as we define it can be found in [15].) Conjecture 1 (Graffiti 747). Let G be a graph. Then D ≤ b 2 . This conjecture has been one of the most circulated of Graffiti’s open conjectures (see [29]). Fajtlowicz was interested in this conjecture in the hope that its proof might result in a more elegant proof of Theorem 2 (the current proofs are rather unwieldy). Note that the following Conjecture 2, which is slightly weaker than Conjecture 1, also generalizes conjecture number 2 of Graffiti. The main results of this paper are some partial resolutions of Conjecture 1 and a near resolution of Conjecture 2. the electronic journal of combinatorics 15 (2008), #R33 2 Conjecture 2. Let G be a graph. Then