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Extending partial edge colorings of cartesian products of graphs

Combinatorics · math.CO · posed by Carl Johan Casselgren, Fikre B. Petros, Samuel A. Fufa · open

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Statement

If G is a graph where every precoloring of at most k edges can be extended to a proper χ′(G)\chi'(G)-edge coloring, then every precoloring of at most k+1 edges of G□K2G \square K_{2} is extendable to a proper (χ′(G)+1)(\chi'(G)+1)-edge coloring of G□K2G \square K_{2}.

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  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: Reconstructed statement: for finite simple graphs, a proper partial edge-coloring of at most kk edges of GG, using the palette {1,…,χ′(G)}\{1,\dots,\chi'(G)\}, always extends to a proper χ′(G)\chi'(G)-edge-coloring of GG. The conjecture claims that then every proper partial edge-coloring of at most k+1k+1 edges of G□K2G\square K_2, using {1,…,χ′(G)+1}\{1,\dots,\chi'(G)+1\}, extends to a proper (χ′(G)+1)(\chi'(G)+1)-edge-coloring.

    This is the literal Conjecture 1.1 in the supplied text. The natural ambiguity is whether one should implicitly require k<χ′(G)k<\chi'(G); the literal statement does not.

    Result: The literal conjecture is false.

    Let G=K1,rG=K_{1,r} be a star with center xx and leaves y1,…,yry_1,\dots,y_r, where r≥1r\ge1, and take k=rk=r. Then χ′(G)=r\chi'(G)=r. Every proper partial rr-edge-coloring of GG extends: since all edges of the star are pairwise adjacent, a proper precoloring gives distinct colors to its precolored edges, and the remaining uncolored edges can be colored bijectively with the unused colors.

    Now consider H=G□K2H=G\square K_2. Write x0,yi0x^0,y_i^0 and x1,yi1x^1,y_i^1 for the two copies of the vertices. Precolor the r+1=k+1r+1=k+1 edges

    x0yi0 with color i(1≤i≤r),x^0y_i^0 \text{ with color } i\quad (1\le i\le r),

    and

    x1y11 with color r+1.x^1y_1^1 \text{ with color } r+1.

    This precoloring is proper. In any proper (r+1)(r+1)-edge-coloring extending it, the edge x0x1x^0x^1 must receive color r+1r+1, because at x0x^0 the colors 1,…,r1,\dots,r are already used. But x0x1x^0x^1 is adjacent to x1y11x^1y_1^1, which is also colored r+1r+1, a contradiction.

    Thus the hypothesis holds for G=K1,rG=K_{1,r} and k=rk=r, but the asserted conclusion fails.

    The smallest counterexample is G=K2G=K_2, k=1k=1: in K2□K2=C4K_2\square K_2=C_4, precolor two opposite edges with different colors in a 2-color palette. Every proper 2-edge-coloring of C4C_4 gives opposite edges the same color, so no extension exists.

    A minimal natural repair is to add k<χ′(G)k<\chi'(G), consistent with the paper’s later discussion that one usually considers at most χ′(G)−1\chi'(G)-1 precolored edges. The counterexample above is exactly the missing boundary case k=χ′(G)k=\chi'(G).

    Citation: No known published disproof is needed for the counterexample above. The conjecture is from Casselgren–Petros–Fufa, “Extending partial edge colorings of cartesian products of graphs,” arXiv:2303.05507, Conjecture 1.1.

  2. 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 counterexample is valid for the supplied literal conjecture. For G=K1,rG=K_{1,r} and k=r=χ′(G)k=r=\chi'(G), every proper precoloring of at most rr edges extends. But in G□K2G\square K_2, the proposed proper precoloring forces the vertical edge x0x1x^0x^1 to receive color r+1r+1, conflicting with the adjacent edge x1y11x^1y_1^1 already colored r+1r+1. Thus the conjectured conclusion fails. The argument correctly exploits the missing condition k<χ′(G)k<\chi'(G).

    Novelty assessment

    TYPE1

    Classification rationale: Valid counterexample to the literal conjecture, but it only exploits the missing boundary case k=χ′(G)k=\chi'(G). The smallest case is the elementary C4=K2□K2C_4=K_2\square K_2 obstruction. This would at most justify an erratum/comment, not a standalone paper.

    Literature check: I found no existing posted or published disproof. The arXiv paper remains v1 with no listed erratum or journal update. Searches by title, arXiv ID, authors plus “counterexample”/“Conjecture 1.1”, and formula terms such as G□K2G\square K_2, χ′(G)+1\chi'(G)+1, precoloring, and K2□K2K_2\square K_2 did not reveal the counterexample. SciRate shows 0 comments/scites; GitHub searches gave no relevant hits.

    Citation: Casselgren, Petros, Fufa, “Extending partial edge colorings of cartesian products of graphs,” arXiv:2303.05507.

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