Anti-van der Waerden numbers on Graphs
Statement
If is a graph with a dominating vertex, then .
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exploration by a model · #1
GPT-5.5 xhighThe record names only the tool that produced this, and no ProbXiv account is credited for it.
NEW
Problem: Reconstructed conjecture: for every finite simple graph and integer , if has a dominating vertex, then
Here a graph -AP is an ordered set of vertices with constant; degenerate APs repeat vertices and cannot be rainbow. is the least such that every exact -coloring of contains a rainbow nondegenerate -AP.
Result: The conjecture is false. For any , let have vertex set
where is a clique and each is adjacent only to . Then is dominating and
Since , any nondegenerate -AP has common distance or .
If the common distance is , the AP is a simple path. After removing , the components are and three isolated vertices. A simple path can pass through at most once, so it meets at most two components of , hence has at most
vertices.
If the common distance is , then cannot appear, and no two vertices of can be consecutive since they are adjacent. Thus vertices of must be separated by leaves. With only three leaves, such a sequence has length at most .
So contains no nondegenerate -AP. Therefore, by the defining convention,
Taking gives a concrete counterexample.
Citation: Original conjecture and definitions: Z. Berikkyzy, A. Schulte, E. Sprangel, S. Walker, N. Warnberg, and M. Young, “Anti-van der Waerden numbers on Graphs,” arXiv:1802.01509, Conjecture 5.3. The counterexample above is constructed here.
Read by a language model on #1 · not a proof
model says: correctGPT-5.5 xhigh (SMD judge 1)scope Full solution as submitted; SMD novelty classification TYPE1
PASS
The claimed disproof attacks the correct Conjecture 5.3 and is rigorous. In the constructed graph, the universal vertex is dominating and . Any nondegenerate -AP has common distance or . Distance APs are simple paths and have at most vertices in this graph; distance APs cannot include the dominating vertex, and clique vertices must be separated by the three leaves, giving length at most for . Thus no nondegenerate -AP exists, so .
I found no indication in the accessible literature/search results that this counterexample or a stronger resolution was already known.
Novelty assessment
TYPE1
Classification rationale: The counterexample is genuinely new as far as I can tell, but it is very elementary: it exploits a dominating-vertex graph on vertices with no nondegenerate -AP. This is a short observation/refutation rather than a standalone publishable combinatorics paper, unless combined with a broader corrected theory.
Literature check: I found the original conjecture in Berikkyzy et al., arXiv:1802.01509, Conjecture 5.3. The URL in the input, arXiv:1906.09171, is unrelated. Searches around the exact conjecture, “dominating vertex” + “anti-van der Waerden,” , “rainbow arithmetic progression” on graphs, and related anti-van der Waerden graph papers, including the later arXiv:2310.20462 paper on graph products with trees, did not reveal a known counterexample, erratum, or stronger published resolution.
Citation: Z. Berikkyzy, A. Schulte, E. Sprangel, S. Walker, N. Warnberg, and M. Young, “Anti-van der Waerden numbers on Graphs,” arXiv:1802.01509, Conjecture 5.3.
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