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# DP color functions versus chromatic polynomials

Citation

Dong, F., & Yang, Y. (2022). DP color functions versus chromatic polynomials. Advances in Applied Mathematics, 134, Article 102301. https://doi.org/10.1016/j.aam.2021.102301

Abstract

For any graph ð, the chromatic polynomial of ð is the function ð (ð, ð) which counts the number of proper m-colorings of ð for each positive integer m. The DP color function ðáŽ
áŽ(ð, ð) of ð, introduced by Kaul and Mudrock in 2019, is a generalization of ð (ð, ð) with ðDP(ð, ð)â€ ð (ð, ð) for each positive integer ð. Let ðáŽ
áŽ(ð)â ð(ð) (resp. ðáŽ
áŽ(ð)< ð(ð)) denote the property that ðáŽ
áŽ(ð, ð)= ð(G, ð) (resp. ðáŽ
áŽ(G, ð)< ð(G, ð)) holds for sufficiently large integers ð. It is an interesting problem of finding graphs ð for which ðáŽ
áŽ(ð)â ð(ð) (resp. ðáŽ
áŽ(ð, ð)< ð(ð, ð)) holds. Kaul and Mudrock showed that if ð has an even girth, then ðáŽ
áŽ(ð)< ð(ð) and Mudrock and Thomason recently proved that ðáŽ
áŽ(ð)â ð(ð) holds for each graph ð which has a dominating vertex. We shall generalize their results in this article. For each edge e in ð, let ð(e)=â if e is a bridge of ð, and let ð(e) be the length of a shortest cycle in ð containing e otherwise. We first show that if ð(e) is even for some edge e in ð, then ðáŽ
áŽ(ð)< ð(ð) holds. However, the converse statement of this conclusion fails with infinitely many counterexamples. We then prove that ðáŽ
áŽ(ð)â ð(ð) holds for every graph G that contains a spanning tree T such that for each eâE(G)âE(T), ð(e) is odd and e is contained in a cycle C of length â(e) with the property that ð(eâ²)< ð(e) for each eâ²âE(C)â(E(T)âª{e}). Some open problems are proposed in this article.

Date Issued

2021

Journal

Advances in Applied Mathematics

DOI

10.1016/j.aam.2021.102301

Grant ID

No. 11971346

Funding Agency

National Natural Science Foundation of China