#### Research Articles

# The size, multipartite Ramsey numbers for C3 versus all graphs up to 4 vertices

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**Authors:**

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Chula Jayawardene ,

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LK

##### About Chula

Department of Mathematics, Faculty of Science, University of Colombo, Colombo 03.

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Lilanthi Samarasekara

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LK

##### About Lilanthi

Department of Mathematics, Faculty of Science, University of Colombo, Colombo 03.

## Abstract

In this paper we restrict our attention to finite graphs containing no loops or multiple edges. The multipartite graph *K*_{j}_{×s }( *j *≥ 3) consisting of *j *partite sets of uniform size *s *is defined as *V *(*K*_{j}_{×s}) = {*v*_{mn }| *m * {1*,*2*,...,j*} and *n *{1*,*2*,...,s*} } and *E *(*K*_{j}_{×s}) = {*v*_{mn }v_{kl }| *m,k * {1*,*2*,...,j*} and *n, l * {1*,*2*,...,s*} where *k *≠ *m*}. The set of vertices of the *m*^{th }partite set is denoted by {*v*_{mn }| *n * {1*,*2*,...,s*}}. If for every two-colouring (red and blue) of the edges of a graph *K*, there exists a copy of *H *in the first colour (red) or a copy of *G *in the second colour (blue), we write *K *→ (*H,G*). Given two simple graphs *H *and *G*, the Ramsey number *r*(*H,G*) is defined as the smallest positive integer *s *such that *K*_{s }→ (*H,G*) and along the same line of reasoning, the multipartite Ramsey number *m*_{j }(*H,G*) is defined as the smallest positive integer *s *such that *K*_{j}_{×s }→ (*H,G*). Thus, multipartite Ramsey number *m*_{j }(*C*_{3}*,G*) is defined as the smallest positive integer *s *such that any red-blue colouring of *K*_{j}_{×s }contains a red *C*_{3 }or a blue *G*. Since only a few multipartite Ramsey numbers for pairs of graphs have been found so far, in this paper we find all such multipartite Ramsey numbers *m*_{j }(*C*_{3}*,G*) when *G *is any graph up to 4 vertices.
**How to Cite: **
Jayawardene, C. & Samarasekara, L., (2017). The size, multipartite Ramsey numbers for C3 versus all graphs up to 4 vertices. Journal of the National Science Foundation of Sri Lanka. 45(1), pp.67–72. DOI: http://doi.org/10.4038/jnsfsr.v45i1.8039
Published on 24 Mar 2017.

Peer Reviewed

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