Non-conformable subgraphs of non-conformable graphs

We show that if G and H are non-conformable graphs, with H being a subgraph of G of the same maximum degree Δ( G), and if Δ(G)⩾⌈ 1 2 |V(G)|⌉ , then | V( H)|=| V( G)|. We also show that this inequality is best possible, for when Δ(G)=⌊ 1 2 |V(G)|⌋ there are examples of graphs G and H with Δ( H)= Δ( G...

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Bibliographic Details
Published in:	Discrete mathematics Vol. 256; no. 1; pp. 203 - 224
Main Authors:	Hilton, A.J.W., Hind, H.R.
Format:	Journal Article
Language:	English
Published:	Amsterdam Elsevier B.V 28-09-2002 Elsevier
Subjects:	Applied sciences Combinatorics Combinatorics. Ordered structures Computer science; control theory; systems Conformability Exact sciences and technology Graph theory Information retrieval. Graph Mathematics Sciences and techniques of general use Theoretical computing Total colouring Total colouring Conformability Subgraph Graph colouring Complete graph Chromatic number
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Summary:	We show that if G and H are non-conformable graphs, with H being a subgraph of G of the same maximum degree Δ( G), and if Δ(G)⩾⌈ 1 2 \|V(G)\|⌉ , then \| V( H)\|=\| V( G)\|. We also show that this inequality is best possible, for when Δ(G)=⌊ 1 2 \|V(G)\|⌋ there are examples of graphs G and H with Δ( H)= Δ( G) and \| V( H)\|<\| V( G)\| which are both non-conformable. We determine all such examples. Interest in this stems from the modified Conformability Conjecture of Chetwynd, Hilton and Hind, which would characterize all graphs G with Δ(G)⩾⌈ 1 2 \|V(G)\|⌉ , for which the total chromatic number χ T( G) satisfies χ T( G)= Δ( G)+1, in terms of non-conformable subgraphs.
ISSN:	0012-365X 1872-681X
DOI:	10.1016/S0012-365X(01)00433-2