### Abstract

A family F of graphs is said to be (δ,Χ)-bounded if there exists a function f(x) satisfying f(x) → ∞ as x → ∞, such that for any graph G from the family, one has f(δ(G)) ≤ Χ(G), where δ(G) and Χ(G) denotes the minimum degree and chromatic number of G, respectively. Also for any set {H_{1},H_{2},... H_{k}} of graphs by Forb(H_{1},H_{2},... H_{k}) we mean the class of graphs that contain no H_{i} as an induced subgraph for any i = 1,k. In this paper we first answer affirmatively the question raised by the second author by showing that for any tree T and positive integer ℓ, Forb(T,K_{ℓ,ℓ}) is a (δ,Χ)-bounded family. Then we obtain a necessary and sufficient condition for Forb(H_{1},H_{2},..., H_{k}) to be a (δ,Χ)-bounded family, where {H_{1},H_{2},. H_{k}} is any given set of graphs. Next we study (δ,Χ)-boundedness of Forb(C) where C is an infinite collection of graphs. We show that for any positive integer ℓ, Forb(K_{ℓ,ℓ},C_{6},C_{8},) is (δ,Χ)-bounded. Finally we show a similar result when C is a collection consisting of unicyclic graphs.

Original language | English |
---|---|

Journal | Electronic Journal of Combinatorics |

Volume | 18 |

Issue number | 1 |

Publication status | Published - 2011 |

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### ASJC Scopus subject areas

- Geometry and Topology
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Electronic Journal of Combinatorics*,

*18*(1).

**On (δ, Χ)-bounded families of graphs.** / Gyárfás, A.; Zaker, Manouchehr.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 18, no. 1.

}

TY - JOUR

T1 - On (δ, Χ)-bounded families of graphs

AU - Gyárfás, A.

AU - Zaker, Manouchehr

PY - 2011

Y1 - 2011

N2 - A family F of graphs is said to be (δ,Χ)-bounded if there exists a function f(x) satisfying f(x) → ∞ as x → ∞, such that for any graph G from the family, one has f(δ(G)) ≤ Χ(G), where δ(G) and Χ(G) denotes the minimum degree and chromatic number of G, respectively. Also for any set {H1,H2,... Hk} of graphs by Forb(H1,H2,... Hk) we mean the class of graphs that contain no Hi as an induced subgraph for any i = 1,k. In this paper we first answer affirmatively the question raised by the second author by showing that for any tree T and positive integer ℓ, Forb(T,Kℓ,ℓ) is a (δ,Χ)-bounded family. Then we obtain a necessary and sufficient condition for Forb(H1,H2,..., Hk) to be a (δ,Χ)-bounded family, where {H1,H2,. Hk} is any given set of graphs. Next we study (δ,Χ)-boundedness of Forb(C) where C is an infinite collection of graphs. We show that for any positive integer ℓ, Forb(Kℓ,ℓ,C6,C8,) is (δ,Χ)-bounded. Finally we show a similar result when C is a collection consisting of unicyclic graphs.

AB - A family F of graphs is said to be (δ,Χ)-bounded if there exists a function f(x) satisfying f(x) → ∞ as x → ∞, such that for any graph G from the family, one has f(δ(G)) ≤ Χ(G), where δ(G) and Χ(G) denotes the minimum degree and chromatic number of G, respectively. Also for any set {H1,H2,... Hk} of graphs by Forb(H1,H2,... Hk) we mean the class of graphs that contain no Hi as an induced subgraph for any i = 1,k. In this paper we first answer affirmatively the question raised by the second author by showing that for any tree T and positive integer ℓ, Forb(T,Kℓ,ℓ) is a (δ,Χ)-bounded family. Then we obtain a necessary and sufficient condition for Forb(H1,H2,..., Hk) to be a (δ,Χ)-bounded family, where {H1,H2,. Hk} is any given set of graphs. Next we study (δ,Χ)-boundedness of Forb(C) where C is an infinite collection of graphs. We show that for any positive integer ℓ, Forb(Kℓ,ℓ,C6,C8,) is (δ,Χ)-bounded. Finally we show a similar result when C is a collection consisting of unicyclic graphs.

UR - http://www.scopus.com/inward/record.url?scp=79957501067&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79957501067&partnerID=8YFLogxK

M3 - Article

VL - 18

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1

ER -