Bounds on Some Ramsey Numbers Involving Quadrilateral

Xu, Xiaodong; Shao, Zehui; P.Radziszowski, Stanistaw

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Ars Combinatoria

Volume 090
Pages: 337-344

Research article

Bounds on Some Ramsey Numbers Involving Quadrilateral

^¹, ^², ^³

¹Guangxi Academy of Sciences Nanning,Guangxi 530007, China

²Department of Control Science and Engineering Huazhong University of Science and Technology Wuhan 430074, China

³Department of Computer Science Rochester Institute of Technology Rochester, NY 14623, USA

Published: 31/01/2009

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Abstract

For graphs $G_{1}, G_{2}, \dots, G_{m}$ , the Ramsey number $R (G_{1}, G_{2}, \dots, G_{m})$ is defined to be the smallest integer $n$ such that any $m$ -coloring of the edges of the complete graph $K_{n}$ must include a monochromatic $G_{i}$ in color $i$ , for some $i$ . In this note, we establish several lower and upper bounds for some Ramsey numbers involving quadrilateral $C_{4}$ , including: $R (C_{4}, K_{9}) \leq 32, 19 \leq R (C_{4}, C_{4}, K_{4}) \leq 22, 31 \leq R (C_{4}, C_{4}, C_{4}, K_{4}) \leq 50, 52 \leq R (C_{4}, K_{4}, K_{4}) \leq 72, 42 \leq R (C_{4}, C_{4}, K_{3}, K_{5}) \leq 76, 87 \leq R (C_{4}, C_{4}, K_{4}, K_{4}) \leq 179.$

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Ars Combinatoria

Bounds on Some Ramsey Numbers Involving Quadrilateral

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