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Ramsey Good Graphs with Long Suspended Paths

Ramsey Good Graphs with Long Suspended Paths A connected graph H with $$|H|\ge \sigma (G)$$ | H | ≥ σ ( G ) is said to be G-good if $$R(G,H)=(\chi (G)-1)(|H|-1)+\sigma (G)$$ R ( G , H ) = ( χ ( G ) - 1 ) ( | H | - 1 ) + σ ( G ) . For an integer $$\ell \ge 3$$ ℓ ≥ 3 , let $$P_\ell $$ P ℓ be a path of order $$\ell $$ ℓ , and $$H^{(\ell )}$$ H ( ℓ ) a graph obtained from H by joining the end vertices of $$P_\ell $$ P ℓ to distinct vertices u, v of H. It is widely known that for any graphs G and H, if $$\ell $$ ℓ is sufficiently large, then $$H^{(\ell )}$$ H ( ℓ ) is G-good. In this note, we show that there exists a constant $$c=c(\Delta )$$ c = c ( Δ ) such that for any graphs G and H with $$\Delta (G)\le \Delta $$ Δ ( G ) ≤ Δ and $$\Delta (H)\le \Delta $$ Δ ( H ) ≤ Δ , if $$\ell \ge c\cdot (|G|+|H|)$$ ℓ ≥ c · ( | G | + | H | ) , then $$H^{(\ell )}$$ H ( ℓ ) is G-good; and if $$n\ge 2\alpha (G)+\Delta ^2(G)+4$$ n ≥ 2 α ( G ) + Δ 2 ( G ) + 4 , then $$P_n$$ P n is G-good. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Graphs and Combinatorics Springer Journals

Ramsey Good Graphs with Long Suspended Paths

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Japan KK, part of Springer Nature
Subject
Mathematics; Combinatorics; Engineering Design
ISSN
0911-0119
eISSN
1435-5914
DOI
10.1007/s00373-018-1910-z
Publisher site
See Article on Publisher Site

Abstract

A connected graph H with $$|H|\ge \sigma (G)$$ | H | ≥ σ ( G ) is said to be G-good if $$R(G,H)=(\chi (G)-1)(|H|-1)+\sigma (G)$$ R ( G , H ) = ( χ ( G ) - 1 ) ( | H | - 1 ) + σ ( G ) . For an integer $$\ell \ge 3$$ ℓ ≥ 3 , let $$P_\ell $$ P ℓ be a path of order $$\ell $$ ℓ , and $$H^{(\ell )}$$ H ( ℓ ) a graph obtained from H by joining the end vertices of $$P_\ell $$ P ℓ to distinct vertices u, v of H. It is widely known that for any graphs G and H, if $$\ell $$ ℓ is sufficiently large, then $$H^{(\ell )}$$ H ( ℓ ) is G-good. In this note, we show that there exists a constant $$c=c(\Delta )$$ c = c ( Δ ) such that for any graphs G and H with $$\Delta (G)\le \Delta $$ Δ ( G ) ≤ Δ and $$\Delta (H)\le \Delta $$ Δ ( H ) ≤ Δ , if $$\ell \ge c\cdot (|G|+|H|)$$ ℓ ≥ c · ( | G | + | H | ) , then $$H^{(\ell )}$$ H ( ℓ ) is G-good; and if $$n\ge 2\alpha (G)+\Delta ^2(G)+4$$ n ≥ 2 α ( G ) + Δ 2 ( G ) + 4 , then $$P_n$$ P n is G-good.

Journal

Graphs and Combinatorics – Springer Journals

Published: Jun 6, 2018

References

  • Ramsey-goodness and otherwise
    Allen, P; Brightwell, G; Skokan, J
  • Ramsey number for cycles in graphs
    Bondy, J; Erdős, P
  • Ramsey numbers involving graphs with long suspended paths
    Burr, SA
  • The Ramsey number of a graph with bounded maximum degree
    Chvátal, V; Rődl, V; Szemerédi, E; Trotter, WT
  • On two problems in graph Ramsey theory
    Conlon, D; Fox, J; Sudakov, B
  • Density theorems for bipartite graphs and related Ramsey-type results
    Fox, J; Sudakov, B
  • On graphs with linear Ramsey numbers
    Graham, RL; Rődl, V; Ruciński, A
  • Ramsey goodness and beyond
    Nikiforov, V; Rousseau, CC
  • Ramsey numbers involving a long path
    Pei, C; Li, Y
  • Calculating Ramsey numbers by partitioning colored graphs
    Pokrovskiy, A
  • The Ramsey numbers for cycles versus wheel of even order
    Zhang, L; Chen, Y; Cheng, TC