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combinatorics / Graph coloring

Kaul-Mudrock Conjecture on the Unlabeled List Color Function

Donner proved in 1992 that the list color function $P_\ell(G,k)$ equals the chromatic polynomial $P(G,k)$ once $k$ is large. Kaul and Mudrock asked whether the analogue holds for Hanlon's unlabeled chromatic polynomial, and could not settle even the edgeless graph, which they posed as a conjecture. The conjecture is true, and it implies that a disconnected graph satisfies the unlabeled analogue of Donner's result whenever all of its components do.

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combinatoricsJul 18, 2026Significance 9/100Registry: site confirmed

Kaul-Mudrock Conjecture on the Unlabeled List Color Function

Prior state unknown→proved

Donner proved in 1992 that the list color function $P_\ell(G,k)$ equals the chromatic polynomial $P(G,k)$ once $k$ is large. Kaul and Mudrock asked whether the analogue holds for Hanlon's unlabeled chromatic polynomial, and could not settle even the edgeless graph, which they posed as a conjecture. The conjecture is true, and it implies that a disconnected graph satisfies the unlabeled analogue of Donner's result…

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Donner proved in 1992 that the list color function $P_\ell(G,k)$ equals the chromatic polynomial $P(G,k)$ once $k$ is large. Kaul and Mudrock asked whether the analogue holds for Hanlon's unlabeled chromatic polynomial, and could not settle even the edgeless graph, which they posed as a conjecture. The conjecture is true, and it implies that a disconnected graph satisfies the unlabeled analogue of Donner's result whenever all of its components do.

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