On path-quasar Ramsey numbers

Binlong Li; Bo Ning

Binlong Li ; Bo Ning

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 68 (2014), / Harvested from The Polish Digital Mathematics Library

Access to full text
Full (PDF)

Résumé

Let $G_{1}$ and $G_{2}$ be two given graphs. The Ramsey number $R (G_{1}, G_{2})$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_{1}$ or $\bar{G}$ contains a $G_{2}$ . Parsons gave a recursive formula to determine the values of $R (P_{n}, K_{1, m})$ , where $P_{n}$ is a path on $n$ vertices and $K_{1, m}$ is a star on $m + 1$ vertices. In this note, we study the Ramsey numbers $R (P_{n}, K_{1} \lor F_{m})$ , where $F_{m}$ is a linear forest on $m$ vertices. We determine the exact values of $R (P_{n}, K_{1} \lor F_{m})$ for the cases $m \leq n$ and $m \geq 2 n$ , and for the case that $F_{m}$ has no odd component. Moreover, we give a lower bound and an upper bound for the case $n + 1 \leq m \leq 2 n - 1$ and $F_{m}$ has at least one odd component.

Publié le : 2014-01-01
EUDML-ID : urn:eudml:doc:289774

@article{bwmeta1.element.ojs-doi-10_17951_a_2014_68_2_11,
     author = {Binlong Li and Bo Ning},
     title = {On path-quasar Ramsey numbers},
     journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
     volume = {68},
     year = {2014},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2014_68_2_11}
}

Binlong Li; Bo Ning. On path-quasar Ramsey numbers. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 68 (2014) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2014_68_2_11/

Bibliographie

Bondy, J. A., Murty, U. S. R., Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976.

Chen, Y., Zhang, Y., Zhang, K., The Ramsey numbers of paths versus wheels, Discrete Math. 290 (1) (2005), 85–87.

Dirac, G. A., Some theorems on abstract graphs, Proc. London. Math. Soc. 2 (1952), 69–81.

Faudree, R. J., Lawrence, S. L., Parsons, T. D., Schelp, R. H., Path-cycle Ramsey numbers, Discrete Math. 10 (2) (1974), 269–277.

Graham, R. L., Rothschild, B. L., Spencer, J. H., Ramsey Theory, Second Edition, John Wiley & Sons Inc., New York, 1990.

Li, B., Ning, B., The Ramsey numbers of paths versus wheels: a complete solution, Electron. J. Combin. 21 (4) (2014), #P4.41.

Parsons, T. D., Path-star Ramsey numbers, J. Combin. Theory, Ser. B 17 (1) (1974), 51–58.

Rousseau, C. C., Sheehan, J., A class of Ramsey problems involving trees, J. London Math. Soc. 2 (3) (1978), 392–396.

Salman, A. N. M., Broersma, H. J., Path-fan Ramsey numbers, Discrete Applied Math. 154 (9) (2006), 1429–1436.

Salman, A. N. M., Broersma, H. J., Path-kipas Ramsey numbers, Discrete Applied Math. 155 (14) (2007), 1878–1884.

Zhang, Y., On Ramsey numbers of short paths versus large wheels, Ars Combin. 89 (2008), 11–20.