# 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

, Volume 34 (4) – Jun 6, 2018
9 pages      /lp/springer_journal/ramsey-good-graphs-with-long-suspended-paths-le9tnnNqhy
Publisher
Springer Journals
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 CombinatoricsSpringer Journals

Published: Jun 6, 2018

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