Min-Li Yu

Gathering algorithms on paths under interference constraints

We study the problem of gathering information from the nodes of a multi-hop radio network into a pre-determined destination node under interference constraints which are modeled by an integer d ≥ 1, so that any node within distance d of a sender cannot receive calls from any other sender. A set of calls which do not interfere with each other is referred to as a round. We give algorithms and lower bounds on the minimum number of rounds for this problem, when the network is a path and the destination node is either at one end or at the center of the path. The algorithms are shown to be optimal for any d in the first case, and for 1 ≤ d ≤ 4, in the second case.

Optimal gathering protocols on paths under interference constraints

We study the problem of gathering information from the nodes of a multi-hop radio network into a predefined destination node under reachability and interference constraints. In such a network, a node is able to send messages to other nodes within reception distance, but doing so it might create interference with other communications. Thus, a message can only be properly received if the receiver is reachable from the sender and there is no interference from another message being transmitted simultaneously. The network is modeled as a graph, where the vertices represent the nodes of the network and the edges, the possible communications. The interference constraint is modeled by a fixed integer d>=1, which implies that nodes within distance d in the graph from one sender cannot receive messages from another node. In this paper, we suppose that each node has one unit-length message to transmit and, furthermore, we suppose that it takes one unit of time (slot) to transmit a unit-length message and during such a slot we can have only calls which do not interfere (called compatible calls). A set of compatible calls is referred to as a round. We give protocols and lower bounds on the minimum number of rounds for the gathering problem when the network is a path and the destination node is either at one end or at the center of the path. These protocols are shown to be optimal for any d in the first case, and for 1@?d@?4, in the second case.

On Hamilton cycle decompositions of the tensor product of complete graphs

In this paper, we show that the tensor product of complete graphs is hamilton cycle decomposable.

Complexity of greedy edge-colouring

The Grundy index of a graph G =(V, E) is the greatest number of colours that the greedy edge-colouring algorithm can use on G. We prove that the problem of determining the Grundy index of a graph G=(V, E) is NP-hard for general graphs. We also show that this problem is polynomial-time solvable for caterpillars. More specifically, we prove that the Grundy index of a caterpillar is Δ(G) or Δ(G)+1 and present a polynomial-time algorithm to determine it exactly.

Corrigendum to (p,1)-total labelling of graphs [Discrete Math. 308 (2008) 496-513]

A (p, 1)-total labelling of a graph G is an assignment of integers to V (G) ∪ E(G) such that: (i) any two adjacent vertices of G receive distinct integers, (ii) any two adjacent edges of G receive distinct integers, and (iii) a vertex and its incident edge receive integers that differ by at least p in absolute value. The span of a (p, 1)-total labelling is themaximum difference between two labels. The minimum span of a (p, 1)-total labelling of G is called the (p, 1)-total number and denoted by λp(G). In [1], Proposition 51 states that if n is an even integer greater than 7, then λn−6(Kn) = 3n− 15. As pointed out by L. Tan and B. Liu, the proof of this proposition was incorrect. We give here a corrected proof of a slightly weaker version of this proposition; the condition n > 11 instead of n > 7 is now required. Proposition 1. If n is an even integer greater than 11, then λn−6(Kn) = 3n− 15. Proof. By Proposition 39 of [1], λn−6(Kn) ≥ 3n− 15. Let i and j be two integers such that i ≤ j. We denote by [i, j] the sets of integers k such that i ≤ k ≤ j. We give an (n−6, 1)-total labelling of Kn in [0, 3n−15] as follows. Label the vertices with {0, 1, 2, 3, 2n−11}∪[2n−9, 3n− 15]. Consider the complete subgraph K induced by the n− 4 vertices labelled in {2n− 11} ∪ [2n− 9, 3n− 15]. Since n− 4 is even, its chromatic index is n− 5. Hence, one can label the edges of K with labels in [0, n− 6] so that two adjacent edges get different labels. Doing so, free to permute the colours, we can make sure that the label n − 6 is not used for the edge (2n−11, 3n−15). For j ∈ [2n−9, 3n−15], label the edge (3, j)with j−n+6, the edge (2, j)with j−n+5, the edge (1, j) with j − n + 4 and the edge (0, j) with j − n + 3. Change the label of (0, 2n − 9) to 3n − 15. Complete the labelling DOI of original article: 10.1016/j.disc.2007.03.034. ∗ Corresponding author. E-mail addresses: fhavet@sophia.inria.fr (F. Havet), joseph.yu@ucfv.ca (M.-L. Yu). 0012-365X/

Optimal Gathering Algorithms in Multi-hop Radio Tree-Networks ith Interferences

– see front matter© 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2010.04.001 2220 F. Havet, M.-L. Yu / Discrete Mathematics 310 (2010) 2219–2220 with the following labelling of the complete subgraph induced by {0, 1, 2, 3, 2n− 11}. 0 1 2 3 2n− 11 0 3n− 18 3n− 17 3n− 16 n− 6 1 3n− 18 3n− 15 2n− 7 3n− 17 2 3n− 17 3n− 15 2n− 8 3n− 16 3 3n− 16 2n− 7 2n− 8 3n− 15 2n− 11 n− 6 3n− 17 3n− 16 3n− 15 References [1] F. Havet, M.-L. Yu, (p, 1)-total labelling of graphs, Discrete Mathematics 308 (2008) 496–513.

IMPROVED BOUNDS FOR GOSSIPING IN MESH BUS NETWORKS

We study the problem of gathering information from the nodes of a multi-hop radio network into a pre-defined destination node under the interference constraints. In such a network, a message can only be properly received if there is no interference from another message being simultaneously transmitted. The network is modeled as a graph, where the vertices represent the nodes and the edges, the possible communications. The interference constraint is modeled by a fixed integer d I ? 1, which implies that nodes within distance d I in the graph from one sender cannot receive messages from another node. In this paper, we suppose that it takes one unit of time (slot) to transmit a unit-length message. A step (or round) consists of a set of non interfering (compatible) calls and uses one slot. We present optimal algorithms that give minimum number of steps (delay) for the gathering problem with buffering possibility, when the network is a tree, the root is the destination and d I = 1. In fact we study the equivalent personalized broadcasting problem instead.

(p,1)-Total labelling of graphs

Improved bounds for the minimum gossiping time in mesh bus networks of arbitrary dimension for 1-port model are given. More precisely, the gossiping protocol consists of steps during which messages are sent via buses and at the end of the protocol, all the nodes should know all the information. Furthermore, during one step a bus can carry at most one message, and each node can either send or receive (not both) on at most one bus The minimum gossiping time of a bus network G is the minimum number of steps required to perform a gossip under this model. Here we determine almost exactly the minimum gossip time for 2-dimensional mesh bus networks and give tight bounds for d-dimensional mesh bus networks.