Taiwanese Journal of Mathematics

STAR MATCHING AND DISTANCE TWO LABELLING

Wensong Lin and Peter Che-Bor Lam

Full-text: Open access

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 subgraphs $K_{1,i}$ of $G$ with $1 \leq i \leq t$. The $t$-star-matching number, denoted by $SM_t(G)$, is the maximum number of vertices covered by a $t$-star-matching of $G$. Clearly $SM_1(G)/2$ is the edge independence number of $G$.

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 $p_v(G)$, is the smallest number of vertex-disjoint paths covering $V(G)$. By $G^c$ we denote the complement graph of $G$.

In this paper, we design a polynomial time algorithm to compute $SM_t(G)$ for any graph $G$ and any integer $t\geq 2$ and studies the properties of $t$-star-matchings of a graph $G$. For any graph $G$, we determine the path covering numbers of $(\mu(G))^c$ and $(G\times \hat{K}_2)^c$ in terms of $SM_4(G^c)$, and the $L'(2,1)$-labelling umbers of $\mu(G)$ and $G\times \hat{K}_2$ in terms of $SM_4(G^c)$, where $\mu(G)$ is the Mycielskian of $G$ and $G\times \hat{K}_2$ is the direct product of $G$ and $\hat{K}_2$ ($\hat{K}_2$ is a graph obtained from $K_2$ by adding a loop on one of its vertices). Our results imply that the path covering numbers of $(\mu(G))^c$ and $(G\times \hat{K}_2)^c$, the $L'(2,1)$-labelling umbers of $\mu(G)$ and $G\times \hat{K}_2$ can be computed in polynomial time for any graph $G$. So, for any graph $G$, it is polynomial-time solvable to determine whether $(\mu(G))^c$ and $(G\times \hat{K}_2)^c$ has a Hamiltonian path. And consequently, for any graph $G=(V,E)$, it is polynomially solvable to determine whether $\lambda(\mu(G))\leq s$ for each $s\geq |V(\mu(G))|$ and $\lambda(G\times \hat{K}_2)\leq s$ for each $s\geq |V(G\times \hat{K}_2)|$. Using these results, we easily determine $L(2,1)$-labelling numbers and $L'(2,1)$-labelling numbers of several classes of graphs..

Article information

Source
Taiwanese J. Math., Volume 13, Number 1 (2009), 211-224.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405279

Digital Object Identifier
doi:10.11650/twjm/1500405279

Mathematical Reviews number (MathSciNet)
MR2489314

Zentralblatt MATH identifier
1173.05016

Subjects
Primary: 05C12: Distance in graphs 05C78: Graph labelling (graceful graphs, bandwidth, etc.) 05C9

Keywords
$L(2,1)$-labelling path covering Hamiltonian path Mycielskian of a graph direct product

Citation

Lin, Wensong; Lam, Peter Che-Bor. STAR MATCHING AND DISTANCE TWO LABELLING. Taiwanese J. Math. 13 (2009), no. 1, 211--224. doi:10.11650/twjm/1500405279. https://projecteuclid.org/euclid.twjm/1500405279


Export citation

References

  • H. L. Bodlaender, T. Kloks, R. B. Tan and J. van Leeuwen, Approximations for l-colorings of graphs, Comput. J., 47 (2004), 193-204.
  • T. Calamoneri, The L$(h,k)$-Labelling Problem: A Survey and Annotated Bibliography, The computer Journal, 49(5) (2006), 585-608.
  • G. J. Chang, L. Huang and X. Zhu, Circular chromatic numbers of Mycielski's graphs, Discrete Math., 205 (1999), 23-37.
  • G. J. Chang and D. Kuo, The $L(2,1)$-labelling Problem on Graphs, SIAM J. Discrete Math., 9 (1996), 309-316.
  • J. Edmonds, Path, trees, and flowers, Canadian J. of Math., 17 (1965), 449-467.
  • G. Fan, Circular chromatic number and Mycielski graphs, Combinatorica, 24(1) (2004), 127-135.
  • J. Fiala, T. Kloks and J. Kratochvl, Fixed-parameter Complexity of $\lambda$-Labelings, Discrete Appl. Math., 113(1) (2001), 59-72.
  • D. C. Fisher, Fractional colorings with large denominators, J. Graph Theory, 20 (1995), 403-409.
  • D. C. Fisher, P. A. McKenna and E. D. Boyer, Biclique parameters of Mycielskians, Congr. Numer., 111 (1995), 136-142.
  • D. C. Fisher, P. A. McKenna and E. D. Boyer, Hamiltionicity, diameter, domination, packing, and biclique partitions of Mycielski's graphs, Disctete Appl. Math., 84 (1998), 93-105.
  • D. A. Fotakis, S. E. Nikoletseas V. G. Papadopoulou and P. G. Spirakis, NP-com- pleteness results and efficient approximations for radiocoloring in planar graphs, In Proc. 25th Int. Symp. on Mathematical Foundations of Computer Science (MFCS 2000), Bratislava, Slovak Republic, 28 August-September, LNCS 1893, pp. 363-372. Springer-Verlag, Berlin, (2000).
  • J. P. Georges, D. W. Mauro and M. A. Whittlesey, Relating path coverings to vertex labellings with a condition at distance Two, Discrete Math., 135 (1994), 103-111.
  • J. R. Griggs and X. T. Jin, Recent progress in mathematics and engineering on optimal graph labellings with distance conditions, J. Combinatorial Optimization, (2006), in press.
  • J. R. Griggs and R. K. Yeh, Labelling graphs with a condition at distance $2$, SIAM J. Discrete Math., 5 (1992), 586-595.
  • H. Hajiabolhassan and X. Zhu, Circular chromatic number and Mycielski construction, J. Graph Theory, 44 (2003), 95-105.
  • P. C. B. Lam, W. Lin, G. Gu and Z. Song, Circular chromatic number and a generalization of construction of Mycielski, J. Combin. Theor. Ser. B., 89 (2003), 195-205.
  • M. Larsen, J. Propp and D. Ullman, The fractional chromatic number of Mycielski's graphs, J. Graph Theory, 19 (1995), 411-416.
  • S. Micali and V. V. Vazirani, An $O(V^{1/2}E)$ algorithm for finding maximum matching in general graphs, Proceedings of the 21st Annual IEEE Symposium on Foundations of Computer Science, 1980, pp. 17-27.
  • J. Mycielski, Sur le coloriage des graphes, Colloq. Math., 3 (1955), 161-162.
  • F. Wang, Z. Song and W. Lin, Group path covering and distance two labelling, manuscript, 2006.
  • R. K. Yeh, Labeling graphs with a condition at distance two, PhD Thesis, University of South Carolina, Columbia, South Carolina, 1990.