Michael Plantholt

The graph reconstruction number

The reconstruction number of graph G is the minimum number of point-deleted subgraphs required in order to uniquely identify the original graph G. We list, based on computer calculations, the reconstruction number for all graphs with at most seven points. Some constructions and conjectures for graphs of higher order are given. the most striking statement is our concluding conjeture that almost all graphs have have reconstruction number three.

Interpolation theorem for diameters of spanning trees

We give an algorithmic approach for transforming any spanning tree of a 2-connected graph into any other spanning tree of the graph. At each step of the transformation we obtain a spanning tree whose diameter differs from that of the previous tree by at most one. Thus if a 2-connected graph G has a as the minimum and b as the maximum diameter of a spanning tree, then for any integer c between a and b , graph G has a spanning tree of diameter c .

Classification of interpolation theorems for spanning trees and other families of spanning subgraphs

We say that a graphical invariant i of a graph interpolates over a family F of graphs if i satisfies the following property: If m and M are the minimum and maximum values (respectively) of i over all graphs in F then for each k, m ⩽ k ⩽ M, there is a graph H in F for which i(H)= k. In previous works it was shown that when F is the set of spanning trees of a connected graph G, a large number of invariants interpolate (some of these invariants require the additional assumption that G be 2-connected). Although the proofs of all these results use the same basic idea of gradually transforming one tree into another via a sequence of edge exchanges, some of these processes require sequences that use more properties of trees than do others. We show that the edge exchange proofs can be divided into three types, in accordance with the extent to which the exchange sequence depends upon properties of spanning trees. This idea is then used to obtain new interpolation results for some invariants, and to show how the exchange methods and interpolation results on spanning trees can be extended to other families of spanning subgraphs.

A sublinear bound on the chromatic index of multigraphs

Abstract The integer round-up φ(G) of the fractional chromatic index yields the standard lower bound for the chromatic index of a multigraph G. We show that if G has even order n, then the chromatic index exceeds φ(G) by at most max{log 3 2 n, 1 + n/30} . More generally, we show that for any real b, 2 3 ⩽ b , the chromatic index of G exceeds φ(G) by at most max{log1/b n, 1 + n(1 − b)/10}. This is used to show that for n sufficiently large, χ(G) ⩽ φ(G) + 1 + √n 1n n/10. Thus the difference between the chromatic index and its lower bound φ(G) is eventually sublinear; that is, for any real c > 0, there exists a positive integer N such that χ(G) − φ(G) N.

The chromatic index of multigraphs of order at most 10

Abstract The maximum of the maximum degree and the “odd set quotients” provides a well-known lower bound φ ( G ) for the chromatic index of a multigraph G . Plantholt proved that if G is a multigraph of order at most 8, its chromatic index equals φ ( G ) and that if G is a multigraph of order 10, the chromatic index of G cannot exceed φ ( G ) + 1. We identify those multigraphs G of order 9 and 10 whose chromatic index equals φ ( G ) + 1, thus completing the determination of the chromatic index of all multigraphs of order at most 10.

The chromatic index of nearly bipartite multigraphs

Abstract We determine the chromatic index of any multigraph which contains a vertex whose detetion results in a bipartite multigraph.

On Decomposing Regular Graphs Into Isomorphic Double-Stars

Abstract A double-star is a tree with exactly two vertices of degree greater than 1. If T is a double-star where the two vertices of degree greater than one have degrees k1+1 and k2+1, then T is denoted by Sk1,k2 . In this note, we show that every double-star with n edges decomposes every 2n-regular graph. We also show that the double-star Sk,k−1 decomposes every 2k-regular graph that contains a perfect matching.

A Combined Logarithmic Bound on the Chromatic Index of Multigraphs

For a multigraph G, the integer round-up of the fractional chromatic index provides a good general lower bound for the chromatic index . For an upper bound, Kahn 1996 showed that for any real there exists a positive integer N so that whenever . We show that for any multigraph G with order n and at least one edge, ). This gives the following natural generalization of Kahns result: for any positive reals , there exists a positive integer N so that + c whenever . We also compare the upper bound found here to other leading upper bounds.

Improved Sufficient Conditions for the Existence of Anti-Directed Hamiltonian Cycles in Digraphs

Let D be a directed graph of order n. An anti-directed (hamiltonian) cycle H in D is a (hamiltonian) cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. In this paper we give sufficient conditions for the existence of anti-directed hamiltonian cycles. Specifically, we prove that a directed graph D of even order n with minimum indegree and outdegree greater than