Non-persistence of equality between chromatic polynomials and list-color functions

Abstract: For any graph $G$, let $P(G,k)$ and $P_{\ell}(G,k)$ denote the chromatic polynomial and the list-color function of $G$, respectively. It remains an open problem whether, for every graph $G$ and integer $k$, the equality $P(G,k)=P_{\ell}(G,k)>0$ implies that $P(G,k+1)=P_{\ell}(G,k+1)$ also holds. In this paper, we answer this question in the negative. For every integer $k\ge 3$, we construct an infinite family of graphs $G$ such that $P(G,k)=P_{\ell}(G,k)>0$ while $P(G,k+1)>P_{\ell}(G,k+1)$. Moreover, using this infinite family of graphs as attachment gadgets, we further show that any graph $H$ with $P(H,k)=P_{\ell}(H,k)>0$ can be developed into an infinite family of graphs $H'$ with $P(H',k)=P_{\ell}(H',k)>0$ and $P(H',k+1)>P_{\ell}(H',k+1)$.

• The paper proves that for every integer k≥3, infinitely many graphs satisfy P(G,k)=Pℓ(G,k)>0 but have P(G,k+1)>Pℓ(G,k+1), disproving persistence.
• The authors construct counterexamples using uniquely k-colorable cores, carefully designed list assignments, and exponential counting estimates that work for all sufficiently large t.
• A vertex-identification gluing inequality propagates non-persistence from any graph with equality at k, while a triangle-free replacement removes dependence on containing a Kₖ clique.

Background and motivation

For a simple graph $G$, the chromatic polynomial $P(G,k)$ counts proper $k$-colorings, while the list-color function $P_\ell(G,k)$ gives the minimum number of $L$-colorings over all $k$-list assignments $L$. Since the constant list assignment $L(v)=[k]$ recovers $P(G,k)$, one always has $P_\ell(G,k)\le P(G,k)$. Donner's deletion–contraction formula shows that equality holds for all sufficiently large $P(G,k)$0 [Donner 1992], and successive bounds on the threshold $P(G,k)$1 were given by Thomassen ($P(G,k)$2), Wang–Qian–Yan, and Dong–Zhang ($P(G,k)$3). A recurring question in this literature—posed repeatedly since Kirov and Naimi's 2016 work—is whether equality is persistent: does $P(G,k)$4 imply $P(G,k)$5? The case $P(G,k)$6 is trivial, and for $P(G,k)$7 Allred and Mudrock characterized all graphs with $P(G,k)$8; each satisfies equality at every $P(G,k)$9. For $k$0, however, the question remained open. This paper answers it negatively.

Main results

The paper establishes two theorems. Theorem 1: for every integer $k$1, there are infinitely many graphs $k$2 such that $k$3 yet $k$4. Theorem 2 strengthens this: any graph $k$5 with $k$6 (for $k$7) can be extended, via a vertex-identification operation, to an infinite family $k$8 exhibiting the same non-persistence. Thus non-persistence is not an isolated phenomenon but can be propagated from any graph where equality holds at level $k$9.

Construction of counterexamples

Fix $P_\ell(G,k)$0. The graph $P_\ell(G,k)$1 consists of three parts: a clique $P_\ell(G,k)$2; two vertices $P_\ell(G,k)$3 where $P_\ell(G,k)$4 is adjacent to all of $P_\ell(G,k)$5 except $P_\ell(G,k)$6 (so $P_\ell(G,k)$7 is uniquely $P_\ell(G,k)$8-colorable, with $P_\ell(G,k)$9 and $L$0 sharing color class $L$1); and $L$2 independent vertices $L$3 ($L$4, $L$5), each adjacent to both $L$6 and $L$7.

The proof proceeds in three steps:

Equality at $L$8. Counting $L$9-colorings in the order $k$0 yields $k$1: the clique forces distinct colors on the $k$2, each $k$3 is then forced to take the color of $k$4, leaving exactly $k$5 choices per $k$6-vertex. The same greedy count works verbatim under any $k$7-list assignment, giving $k$8, hence equality.

Lower bound at $k$9. Coloring $L$0 identically first gives $L$1.

Upper bound via a crafted $L$2-assignment. The key gadget is a $L$3-list assignment built from a common palette $L$4 of $L$5 colors plus two private pairs $L$6: the $L$7 receive $L$8, $L$9 receives $L(v)=[k]$0, and each $L(v)=[k]$1 receives $L(v)=[k]$2 where the $L(v)=[k]$3 run through the four two-element combinations of $L(v)=[k]$4. An exact enumeration of extensions of colorings of $L(v)=[k]$5 (Lemma on extension counts, distinguishing whether $L(v)=[k]$6 and how many of the two colors lie in $L(v)=[k]$7) yields a closed-form expression for $L(v)=[k]$8. Comparing with the lower bound reduces to showing that

$L(v)=[k]$9

which holds for all $P(G,k)$0 by a standard exponential-decay estimate. Hence $P(G,k)$1 for all sufficiently large $P(G,k)$2, proving Theorem 1. The mechanism is structural: the dominant term $P(G,k)$3 in $P(G,k)$4 comes from colorings with $P(G,k)$5, while the ordinary polynomial admits many more colorings once $P(G,k)$6 is large enough to amplify the exponential gap between base rates $P(G,k)$7 and the smaller bases $P(G,k)$8, $P(G,k)$9, and $P_\ell(G,k)\le P(G,k)$0.

Propagation via vertex-gluings

Theorem 2 rests on a gluing inequality proved by an averaging argument over permutations of the identified vertex's palette: for any vertex-disjoint $P_\ell(G,k)\le P(G,k)$1,

$P_\ell(G,k)\le P(G,k)$2

The authors note this fails for $P_\ell(G,k)\le P(G,k)$3 gluings (e.g., $P_\ell(G,k)\le P(G,k)$4 at $P_\ell(G,k)\le P(G,k)$5), so only the 1-sum is used. Given any $P_\ell(G,k)\le P(G,k)$6 with $P_\ell(G,k)\le P(G,k)$7, form $P_\ell(G,k)\le P(G,k)$8 by identifying a vertex of $P_\ell(G,k)\le P(G,k)$9 with $P(G,k)$00. At level $P(G,k)$01, the chromatic product formula gives $P(G,k)$02, and a matching lower bound for $P(G,k)$03 follows because every coloring of $P(G,k)$04 extends greedily to at least $P(G,k)$05 colorings of $P(G,k)$06. At level $P(G,k)$07, combining the gluing inequality with $P(G,k)$08 gives strict separation. This mirrors earlier constructions of non-chromatic-adherent DP-color functions via generalized theta graphs, but here the propagation is universal: no hypothesis beyond equality at level $P(G,k)$09 is needed.

Eliminating clique dependence

All graphs in the main construction contain $P(G,k)$10. The concluding section removes this artifact using $P(G,k)$11, the triangle-free uniquely 3-colorable graph of Akbari, Mirrokni, and Sadjad ($P(G,k)$12, $P(G,k)$13). By Akbari–Mirrokni–Sadjad's theorem relating uniquely list-colorable graphs to size-based list assignments, together with a direct argument producing six $P(G,k)$14-colorings for every 3-list assignment, the paper proves $P(G,k)$15. Then $P(G,k)$16—which contains no $P(G,k)$17—is uniquely $P(G,k)$18-colorable with $P(G,k)$19, using the join bound $P(G,k)$20. Replacing $P(G,k)$21 by $P(G,k)$22 produces a second infinite family of counterexamples free of $P(G,k)$23, via proofs analogous to those above.

Limitations and open questions

The paper leaves several points open. First, the counterexamples require $P(G,k)$24 with $P(G,k)$25 growing as roughly $P(G,k)$26; the minimal threshold for each fixed $P(G,k)$27 is not determined. Second, the characterization question remains unresolved: which graphs $P(G,k)$28 have the property that $P(G,k)$29 implies persistence at $P(G,k)$30 for all $P(G,k)$31? The $P(G,k)$32 classification of Allred and Mudrock suggests such a characterization may be tractable, but no general criterion is offered. Third, the analogous persistence question for the DP-color function was already answered negatively, and the relationship between the DP and list versions of these phenomena is not explored here. Finally, the gluing inequality is established only for 1-sums; whether some corrected form holds for higher-order gluings is left aside after the $P(G,k)$33 counterexample.

Conclusion

This paper settles a decade-old open question by constructing, for each $P(G,k)$34, infinite families of graphs where equality between the chromatic polynomial and the list-color function holds at $P(G,k)$35 but fails at $P(G,k)$36. The construction combines a uniquely $P(G,k)$37-colorable core with a carefully designed list assignment whose deficit grows exponentially in a tunable parameter, and a permutation-averaging gluing lemma extends the phenomenon to arbitrary seed graphs. The result shows that verifying $P(G,k)$38 at a single value of $P(G,k)$39 provides no guarantee at larger values, sharpening the picture of when the list-color function agrees with the chromatic polynomial.