| Title | Star matching and distance two labelling |
| Creator | |
| Date Issued | 2009 |
| Source Publication | Taiwanese Journal of Mathematics
 |
| ISSN | 1027-5487 |
| Volume | 13Issue:1Pages:211-224 |
| Abstract | Abstract. This paper first introduces a new graph parameter. Let t be a positive integer. A t-star-matching of a graph G is a collection of mutually vertex disjoint subgraphsK ofGwith 1≤i≤t. The t-star-matching number, denoted by SM(G), is the maximum number of vertices covered by at-star- matching ofG. Clearly SM(G)/2 is the edge independence number ofG. An L(2, 1)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that vertices at distance at most two get different numbers and adjacent vertices get numbers which are at least two apart. The L(2, 1)-labelling number of a graph G is the minimum range of labels over all L(2, 1)-labellings. If we require the assignment to be one-to-one, then similarly as above we can define the L′(2, 1)-labelling and the L′(2, 1)-labelling number of a graph G. Given a graph G, the path covering number of G, denoted by p(G), is the smallest number of vertex-disjoint paths covering V (G). By G we denote the complement graph of G. In this paper, we design a polynomial time algorithm to compute SM(G) for any graph G and any integer t ≥ 2 and studies the properties of t-star- matchings of a graph G. For any graph G, we determine the path covering numbers of (μ(G)) and (G × K) in terms of SM(G), and the L′(2, 1)- labelling umbers of μ(G) and G × K in terms of SM(G), where μ(G) is the Mycielskian of G and G× K is the direct product of G and K (2 isa graph obtained from K by adding a loop on one of its vertices). Our results imply that the path covering numbers of (μ(G)) and (G× K), the L′(2, 1)- labelling umbers of μ(G) and G× K can be computed in polynomial time for any graph G. So, for any graph G, it is polynomial-time solvable to determine whether (μ(G)) and (G × K) has a Hamiltonian path. And consequently, for any graph G = (V, E), it is polynomially solvable to determine whether λ(μ(G)) ≤ s for each s ≥ |V (μ(G))| and λ(G× K) ≤ s for each s ≥ |V (G× K)|. Using these results, we easily determine L(2, 1)-labelling numbers and L′(2, 1)-labelling numbers of several classes of graphs. |
| Keyword | Direct product Hamiltonian path L(2,1)-labelling Mycielskian of a graph Path covering |
| DOI | 10.11650/twjm/1500405279 |
| URL | View source |
| Language | 英语English |
| Scopus ID | 2-s2.0-74049134973 |
| Citation statistics | |
| Document Type | Journal article |
| Identifier | http://repository.uic.edu.cn/handle/39GCC9TT/6648 |
| Collection | Research outside affiliated institution |
| Corresponding Author | Lin，Wensong |
| Affiliation | 1.Department of Mathematics,Southeast University,Nanjing 210096,China 2.HKBU-BNU International College,Zhuhai,China 3.Tunghai University,Taichung,Taiwan |
| Recommended Citation GB/T 7714 | Lin，Wensong,Lam，Peter Che Bor. Star matching and distance two labelling[J]. Taiwanese Journal of Mathematics, 2009, 13(1): 211-224. |
| APA | Lin，Wensong, & Lam，Peter Che Bor. (2009). Star matching and distance two labelling. Taiwanese Journal of Mathematics, 13(1), 211-224. |
| MLA | Lin，Wensong,et al."Star matching and distance two labelling". Taiwanese Journal of Mathematics 13.1(2009): 211-224. |
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