DP color functions versus chromatic polynomials

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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
Author
Dong, F. M. 
•
Yang, Yan
Abstract
For any graph 𝘎, the chromatic polynomial of 𝘎 is the function 𝘗 (𝘎, 𝓂) which counts the number of proper m-colorings of 𝘎 for each positive integer m. The DP color function 𝘗ᎅ᎘(𝘎, 𝓂) of 𝘎, introduced by Kaul and Mudrock in 2019, is a generalization of 𝘗 (𝘎, 𝓂) with 𝘗DP(𝘎, 𝓂)≀ 𝘗 (𝘎, 𝓂) for each positive integer 𝓂. Let 𝘗ᎅ᎘(𝘎)≈ 𝘗(𝘎) (resp. 𝘗ᎅ᎘(𝘎)< 𝘗(𝘎)) denote the property that 𝘗ᎅ᎘(𝘎, 𝓂)= 𝘗(G, 𝓂) (resp. 𝘗ᎅ᎘(G, 𝓂)< 𝘗(G, 𝓂)) holds for sufficiently large integers 𝓂. It is an interesting problem of finding graphs 𝘎 for which 𝘗ᎅ᎘(𝘎)≈ 𝘗(𝘎) (resp. 𝘗ᎅ᎘(𝘎, 𝓂)< 𝘗(𝘎, 𝓂)) holds. Kaul and Mudrock showed that if 𝘎 has an even girth, then 𝘗ᎅ᎘(𝘎)< 𝘗(𝘎) and Mudrock and Thomason recently proved that 𝘗ᎅ᎘(𝘎)≈ 𝘗(𝘎) holds for each graph 𝘎 which has a dominating vertex. We shall generalize their results in this article. For each edge e in 𝘎, let 𝓁(e)=∞ if e is a bridge of 𝘎, and let 𝓁(e) be the length of a shortest cycle in 𝘎 containing e otherwise. We first show that if 𝓁(e) is even for some edge e in 𝘎, then 𝘗ᎅ᎘(𝘎)< 𝘗(𝘎) holds. However, the converse statement of this conclusion fails with infinitely many counterexamples. We then prove that 𝘗ᎅ᎘(𝘎)≈ 𝘗(𝘎) holds for every graph G that contains a spanning tree T such that for each e∈E(G)∖E(T), 𝓁(e) is odd and e is contained in a cycle C of length ℓ(e) with the property that 𝓁(e′)< 𝓁(e) for each e′∈E(C)∖(E(T)∪{e}). Some open problems are proposed in this article.
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Date Issued
2021
Publisher
Elsevier
Journal
Advances in Applied Mathematics
DOI
10.1016/j.aam.2021.102301
Grant ID
No. 11971346
Funding Agency
National Natural Science Foundation of China