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If the graph G contains a cycle C of length m; but G does not contain any cycle of length m + 1; and the complement of G has no Kr; every vertex of V \ V is adjacent to at most r − 2 vertices of C

# On book-wheel Ramsey number

Discrete Mathematics, no. 1-3 (2000): 239-249

Cited by: 1|Views21
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Abstract

In this paper we determine the following Ramsey numbers: (1) r ( B m , W n )=2 n +1 for m⩾1, n⩾5m+3 , (2) r ( B m , K 2 ∨ C n )=2 n +3 for n ⩾9 if m =1 or n ⩾( m −1)(16 m 3 +16 m 2 −24 m −10)+1 if m ⩾2, where the book B m is the join K 2 ∨ K m c , W n denotes a wheel with n spokes, and C n denotes a cycle of length n.

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Introduction
• If the edges of the complete graph are colored either red or blue, denote the spanning subgraph with all red edges and all blue edges, respectively, by R and B.
• In this case the authors obtain |V1 ∪ T1|¿|V (B) \ (V2 ∪ T2 ∪ {u; v})| − 1¿2n + 3 − (n − 1 + 2) − 1 = n + 1 and the authors have the B[V1] is a complete graph, since |NB(V1) ∩ V2|61 and R + Bm. if |V1|¿4, every edge of B[V1] is required.
Highlights
• If the edges of the complete graph are colored either red or blue, denote the spanning subgraph with all red edges and all blue edges, respectively, by R and B
• If the graph G contains a cycle C of length m; but G does not contain any cycle of length m + 1; and the complement of G has no Kr; every vertex of V (G) \ V (C) is adjacent to at most r − 2 vertices of C
• The closure C(G) of a graph G is the graph obtained from G by recursively joining pairs of nonadjacent vertices whose degree sum is at least |V (G)| until no such pair remains
• Applying Lemma 5 to the graph B[V2 ∪ V3 \ T2], we may assume that xy is an edge of B[V2] satisfying |NB(x) ∩ NB(y) ∩ (V2 ∪ V3 \ T2)|¿|V2| + |V3 \ T2|=4 − 11m=4 + 3=4
• We only prove the ÿrst part of the fact; the proof of the second is the same
• If R + Bm and B + K2 ∨ Cn, by Proposition 2.1, we may assume that uv is an edge in B such that B[NB(u) ∩ NB(v)] contains a block G with at least n vertices
Results
• If |D|63m + 2, |V2 \ D| − (m − 1)2 − m¿|NB(u) ∩ NB(v)| − |V1| − 1 − |D| − (m − 1)2 − m¿n − 3 − 1 − (3m + 2) − (m − 1)2 − m ¿ 0 and the authors have that (B[V2])¿|V2| − m; |V1| ¡ m since |NB(V1) ∩ V2|61 and R + Bm. Applying Lemma 5 to the graph B[V2 ∪ V3 \ T2] the authors may assume that xy is an edge of B[V2] satisfying |NB(x) ∩ NB(y) ∩ (V2 ∪ V3 \ T2)|¿|V2| + |V3 \ T2|=4 − 11m=4 + 3=4.
• Applying Lemma 5 to the graph B[V2 ∪ T1 \ T2], the authors may assume that xy is an edge of B[V2] such that |NB(x) ∩ NB(y) ∩ (V2 ∪ T1 \ T2)|¿|V2| + |T1 \ T2|=4−11m=4+3=4.
• Observing that |V2 ∪ T2|¿n−|V3 \ (T1 ∪ T2)|¿n−7m−2 and |T1 \ T2|=|V (B) \ (V2 ∪ T2 ∪ V (H1) ∪ {u; v}|−|V3 \ (T1 ∪ T2)|¿2n+3−|V2 ∪ T2|−m−2− (7m+2)=2n−|V2 ∪ T2|−8m−1, the authors obtain |NB(x)∩NB(y)|¿|V2|+|T1 \ T2|=4−11m=4+ 3=4+|T2|+|{u; v}|¿|V2 ∪ T2|+(2n−|V2 ∪ T2|−8m−1)=4−11m=4+11=4=3|V2 ∪ T2|=4+ n=2 − 19m=4 + 5=2¿3(n − 7m − 2)=4 + n=2 − 19m=4 + 5=2 ¿ n + m ¿ |NB(u) ∩ NB(v)|, contradicting the choice of the edge uv.
• B[NB(x)∩T1 \ T2] contains a required edge since |V1 ∪ T1 \ T2|¿n + 1 and B[V1 ∪ T1 \ T2] is a complete graph.
Conclusion
• Applying Lemma 5 to the graph B[V2 ∪ T1] and observing that |V2|¿n − 3 and |T1|¿n − 2, the authors can ÿnd an edge xy in B[V2] such that |NB(x) ∩ NB(y) ∩ (V2 ∪ T1)|¿|V2| + |T1|=4 − 11m=4 + 3=4¿n − 3 + (n − 2)=4 − 11m=4 + 3=4 ¿ n + m ¿ |NB(u) ∩ NB(v)|, a contradiction.
• If R + Bm and B + K2 ∨ Cn, by Proposition 2.1, the authors may assume that uv is an edge in B such that B[NB(u) ∩ NB(v)] contains a block G with at least n vertices.
Reference
• J.A. Bondy, U.S.R. Murty, Graph Theory With Applications, Macmilan, London and Elsevier, New York, 1976.
• J.A. Bondy, A.W. Ingleton, Pancyclic Graph, J. Combin. Theory B 80 (1976) 41–46.
• S.A. Burr, P. Erdos, Generalizations of a Ramsey — theoretic result of Chvatal, J. Graph Theory 7 (1983) 39–51.
• G.A. Dirac, Hamilton circuits and long circuits, Ann. Discrete Math. 3 (1978) 75–92.
• J. Sheehan, Finite Ramsey theory and strongly regular graph, Ann. Discrete Math. 13 (1982) 179–190.
• H.-L. Zhou, On Ramsey numbers for odd cycle versus wheel, J. Math. (PRC) 15 (1995) 119–120.
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