Some three-color Ramsey numbers, R(P4,P5,Ck) and R(P4,P6,Ck)

  • Authors:

  • Zehui Shao;Xiaodong Xu;Xiaolong Shi;Linqiang Pan

  • Affiliations:

  • Key Laboratory of Image Processing and Intelligent Control, Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;Guangxi Academy of Science, Nanning, Guangxi 530007, China;Key Laboratory of Image Processing and Intelligent Control, Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;Key Laboratory of Image Processing and Intelligent Control, Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2009

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Abstract

For given graphs G"1,G"2,G"3, the three-color Ramsey number R(G"1,G"2,G"3) is defined to be the least positive integer n such that every 3-coloring of the edges of complete graph K"n contains a monochromatic copy of G"i colored with i, for some 1@?i@?3. In this paper, we prove that R(P"4,P"5,C"3)=11, R(P"4,P"5,C"4)=7, R(P"4,P"5,C"5)=11, R(P"4,P"5,C"7)=11, R(P"4,P"5,C"k)=k+2 for k=23; R(P"4,P"6,C"4)=8, R(P"4,P"6,C"3)=R(P"4,P"6,C"5)=R(P"4,P"6,C"7)=13, R(P"4,P"6,C"k)=k+3 for k=18.