Geodesic
On the Ramsey numbers of non-star trees versus connected graphs of order six
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 2, pp. 331-349

Voir la notice de l'article provenant de la source Library of Science

This paper completes our studies on the Ramsey number r(T_n,G) for trees T_n of order n and connected graphs G of order six. If χ(G) ≥ 4, then the values of r(T_n,G) are already known for any tree T_n. Moreover, r(S_n,G), where S_n denotes the star of order n, has been investigated in case of χ(G) ≤ 3. If χ(G) = 3 and G K_2,2,2, then r(S_n,G) has been determined except for some G and some small n. Partial results have been obtained for r(S_n,K_2,2,2) and for r(S_n,G) with χ(G) = 2. In the present paper we investigate r(T_n,G) for non-star trees T_n and χ(G) ≤ 3. Especially, r(T_n,G) is completely evaluated for any non-star tree T_n if χ(G) = 3 where G K_2,2,2, and r(T_n,K_2,2,2) is determined for a class of trees T_n with small maximum degree. In case of χ(G) = 2, r(T_n,G) is investigated for T_n = P_n, the path of order n, and for T_n= B_2,n-2, the special broom of order n obtained by identifying the centre of a star S_3 with an end-vertex of a path P_n-2. Furthermore, the values of r(B_2,n-2,S_m) are determined for all n and m with n ≥ m - 1. As a consequence of this paper, r(F,G) is known for all trees F of order at most five and all connected graphs G of order at most six.
Keywords: Ramsey number, Ramsey goodness, tree, star, path, broom, small graph 
@article{DMGT_2023_43_2_a2,
     author = {Lortz, Roland and Mengersen, Ingrid},
     title = {On the {Ramsey} numbers of non-star trees versus connected graphs of order six},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {331--349},
     publisher = {mathdoc},
     volume = {43},
     number = {2},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2023_43_2_a2/}
}
TY  - JOUR
AU  - Lortz, Roland
AU  - Mengersen, Ingrid
TI  - On the Ramsey numbers of non-star trees versus connected graphs of order six
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2023
SP  - 331
EP  - 349
VL  - 43
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2023_43_2_a2/
LA  - en
ID  - DMGT_2023_43_2_a2
ER  - 
%0 Journal Article
%A Lortz, Roland
%A Mengersen, Ingrid
%T On the Ramsey numbers of non-star trees versus connected graphs of order six
%J Discussiones Mathematicae. Graph Theory
%D 2023
%P 331-349
%V 43
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2023_43_2_a2/
%G en
%F DMGT_2023_43_2_a2
Lortz, Roland; Mengersen, Ingrid. On the Ramsey numbers of non-star trees versus connected graphs of order six. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 2, pp. 331-349. http://geodesic.mathdoc.fr/item/DMGT_2023_43_2_a2/