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On Ramsey numbers for paths versus wheels
Salman A.N.M.a, Broersma H.J.b
a Combinatorial Mathematics Research Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia
b Department of Computer Science, University of Durham, United Kingdom
[vc_row][vc_column][vc_row_inner][vc_column_inner][vc_separator css=”.vc_custom_1624529070653{padding-top: 30px !important;padding-bottom: 30px !important;}”][/vc_column_inner][/vc_row_inner][vc_row_inner layout=”boxed”][vc_column_inner width=”3/4″ css=”.vc_custom_1624695412187{border-right-width: 1px !important;border-right-color: #dddddd !important;border-right-style: solid !important;border-radius: 1px !important;}”][vc_empty_space][megatron_heading title=”Abstract” size=”size-sm” text_align=”text-left”][vc_column_text]For two given graphs F and H, the Ramsey number R (F, H) is the smallest positive integer p such that for every graph G on p vertices the following holds: either G contains F as a subgraph or the complement of G contains H as a subgraph. In this paper, we study the Ramsey numbers R (Pn, Wm), where Pn is a path on n vertices and Wm is the graph obtained from a cycle on m vertices by adding a new vertex and edges joining it to all the vertices of the cycle. We present the exact values of R (Pn, Wm) for the following values of n and m: n = 1, 2, 3 or 5 and m ≥ 3; n = 4 and m = 3, 4, 5 or 7; n ≥ 6 and (m is odd, 3 ≤ m ≤ 2 n – 1) or (m is even, 4 ≤ m ≤ n + 1); odd n ≥ 7 and m = 2 n – 2 or m = 2 n or m ≥ (n – 3)2; odd n ≥ 9 and q · n – 2 q + 1 ≤ m ≤ q · n – q + 2 with 3 ≤ q ≤ n – 5. Moreover, we give nontrivial lower bounds and upper bounds for R (Pn, Wm) for the other values of m and n. © 2006 Elsevier B.V. All rights reserved.[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Author keywords” size=”size-sm” text_align=”text-left”][vc_column_text]Nontrivial lower bounds,Positive integer,Ramsey number,Subgraph[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Indexed keywords” size=”size-sm” text_align=”text-left”][vc_column_text]Path,Ramsey number,Wheel[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Funding details” size=”size-sm” text_align=”text-left”][vc_column_text][/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”DOI” size=”size-sm” text_align=”text-left”][vc_column_text]https://doi.org/10.1016/j.disc.2005.11.049[/vc_column_text][/vc_column_inner][vc_column_inner width=”1/4″][vc_column_text]Widget Plumx[/vc_column_text][/vc_column_inner][/vc_row_inner][/vc_column][/vc_row][vc_row][vc_column][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][/vc_column][/vc_row]