On size multipartite Ramsey numbers for stars

Abstract

<p>Burger and Vuuren defined the size multipartite Ramsey number for a pair of complete, balanced, multipartite graphs <em>mj</em>(<em>Ka</em>x<em>b</em>,<em>Kc</em>x<em>d</em>), for natural numbers <em>a,b,c,d</em> and <em>j</em>, where <em>a,c</em> >= 2, in 2004. They have also determined the necessary and sufficient conditions for the existence of size multipartite Ramsey numbers <em>mj</em>(<em>Ka</em>x<em>b</em>,<em>Kc</em>x<em>d</em>). Syafrizal <em>et al</em>. generalized this definition by removing the completeness requirement. For simple graphs <em>G</em> and <em>H</em>, they defined the size multipartite Ramsey number <em>mj</em>(<em>G,H</em>) as the smallest natural number <em>t</em> such that any red-blue coloring on the edges of <em>Kj</em>x<em>t</em> contains a red <em>G</em> or a blue <em>H</em> as a subgraph. In this paper, we determine the necessary and sufficient conditions for the existence of multipartite Ramsey numbers <em>mj</em>(<em>G,H</em>), where both <em>G</em> and <em>H</em> are non complete graphs. Furthermore, we determine the exact values of the size multipartite Ramsey numbers <em>mj</em>(<em>K</em>1,<em>m</em>, <em>K</em>1,<em>n</em>) for all integers <em>m,n >= </em>1 and <em>j </em>= 2,3, where <em>K</em>1,<em>m</em> is a star of order <em>m</em>+1. In addition, we also determine the lower bound of <em>m</em>3(<em>kK</em>1,<em>m</em>, <em>C</em>3), where <em>kK</em>1,<em>m</em> is a disjoint union of <em>k</em> copies of a star <em>K</em>1,<em>m</em> and <em>C</em>3 is a cycle of order 3.</p>

The Size, Multipartite Ramsey Numbers for nK2 Versus Path–Path and Cycle

“…Recently, the numbers m j (G 1 , G 2 ) have been investigated for special classes: stripes versus cycles; and stars versus cycles, see [10] and its references. In [15], authors determined the necessary and sufficient conditions for the existence of multipartite Ramsey numbers m j (G, H) where both G and H are incomplete graphs, which also determined the exact values of the size multipartite Ramsey numbers m j (K 1,m , K 1,n ) for all integers m, n ≥ 1 and j = 2, 3. Syafrizal et al determined the size multipartite Ramsey numbers of path versus path [16].…”