Li-Da Tong

Geodetic spectra of graphs

Abstract Geodetic numbers of graphs and digraphs have been investigated in the literature recently. The main purpose of this paper is to study the geodetic spectrum of a graph. For any two vertices u and v in an oriented graph D , a u – v geodesic is a shortest directed path from u to v . Let I ( u , v ) denote the set of all vertices lying on a u – v geodesic. For a vertex subset A , let I ( A ) denote the union of all I ( u , v ) for u , v ∈ A . The geodetic number g ( D ) of an oriented graph D is the minimum cardinality of a set A with I ( A )= V ( D ). The (strong) geodetic spectrum of a graph G is the set of geodetic numbers of all (strongly connected) orientations of G . In this paper, we determine geodetic spectra and strong geodetic spectra of several classes of graphs. A conjecture and two problems given by Chartrand and Zhang are dealt with.

The hamiltonian property of the consecutive-3 digraph

A consecutive-ifd digraph is a digraph G(d, n, q, r) whose n nodes are labeled by the residues modulo n and a link from node i to node j exists if and only if j @? qi + k (mod n) for some k with r @? k @? r + d - 1. Consecutive-d digraphs are used as models for many computer networks and multiprocessor systems, in which the existence of a Hamiltonian circuit is important. Conditions for a consecutive-d graph to have a Hamiltonian circuit were known except for gcd(n, d) = 1 and d = 3 or 4. It was conjectured by Du, Hsu, and Hwang that a consecutive-3 digraph is Hamiltonian. This paper produces several infinite classes of consecutive-3 digraphs which are not (respectively, are) Hamiltonian, thus suggesting that the conjecture needs modification.

Strictly nonblocking three-stage Clos networks with some rearrangeable multicast capability

F.K. Hwang and S.C. Liaw (see IEEE/ACM Trans. Networking, vol.8, p.535-9, 2000) introduced a new nonblocking requirement for 2-cast traffic which imposes different requirements on different types of coexisting calls. The requirement is strictly nonblocking for point-to-point calls among the 2-cast traffic, and is rearrangeable for genuine 2-cast calls. We generalize the 2-cast calls to multicast calls and give a sufficient condition for such networks when the number of multicast calls is upper bounded.

A note on the Gallai---Roy---Vitaver theorem

The well-known theorem by Gallai-Roy-Vitaver says that every digraph G has a directed path with at least χ(G) vertices; hence this holds also for graphs. Li strengthened the digraph result by showing that the directed path can be constrained to start from any vertex that can reach all others. For a graph G given a proper χ(G)-coloring, he proved that the path can be required to start at any vertex and visit vertices of all colors. We give a shorter proof of this. He conjectured that the same holds for digraphs; we provide a strongly connected counterexample. We also give another extension of the Gallai-Roy-Vitaver Theorem on graphs.

Channel graphs of bit permutation networks

Channel graphs have been widely used in the study of blocking networks. In this paper, we show that a bit permutation network has a unique channel graph if and only if it is connected, and two connected bit permutation networks are isomorphic if and only if their channel graphs are isomorphic.

Hamiltonian numbers of Möbius double loop networks

For the study of hamiltonicity of graphs and digraphs, Goodman and Hedetniemi introduced the concept of Hamiltonian number. The Hamiltonian numberh(D) of a digraph D is the minimum length of a closed walk containing all vertices of D. In this paper, we study Hamiltonian numbers of the following proposed networks, which include strongly connected double loop networks. For integers d≥1, m≥1 and ℓ≥0, the Möbius double loop network MDL(d,m,ℓ) is the digraph with vertex set {(i,j):0≤i≤d−1,0≤j≤m−1} and arc set {(i,j)(i+1,j) or (i,j)(i+1,j+1):0≤i≤d−2,0≤j≤m−1}∪{(d−1,j)(0,j+ℓ) or (d−1,j)(0,j+ℓ+1):0≤j≤m−1}, where the second coordinate y of a vertex (x,y) is taken modulo m. We give an upper bound for the Hamiltonian number of a Möbius double loop network. We also give a necessary and sufficient condition for a Möbius double loop network MDL(d,m,ℓ) to have Hamiltonian number at most dm, dm+d, dm+1 or dm+2.

Hamiltonian numbers in oriented graphs

A hamiltonian walk of a digraph is a closed spanning directed walk with minimum length in the digraph. The length of a hamiltonian walk in a digraph D is called the hamiltonian number of D, denoted by h(D). In Chang and Tong (J Comb Optim 25:694–701, 2013), Chang and Tong proved that for a strongly connected digraph D of order n,

Independent arcs of acyclic orientations of complete r-partite graphs

n\le h(D)\le \lfloor \frac{(n+1)^2}{4} \rfloor