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DP color functions versus chromatic polynomials (II)
Citation
Zhang, M., & Dong, F. (2023). DP color functions versus chromatic polynomials (II). Journal of Graph Theory, 103(4), 740-761. https://doi.org/10.1002/jgt.22944
Abstract
For any connected graph G, let P(G,m) and PDP (G,m) denote the chromatic polynomial and DP color function of G, respectively. It is known that PDP (G,m) ≤ P(G,m) holds for every positive integer m. Let DP≈ (resp. DP<) be the set of graphs G for which there exists an integer M such that PDP (G,m) = P(G,m) (resp. PDP (G,m) < P(G,m)) holds for all integers m ≥ M. Determining the sets DP≈ and DP< is an important open problem on the DP color function. For any edge set E0 of G, let ℓG(E0) be the size of a shortest cycle C in G such that |E(C) ∩ E0| is odd if such a cycle exists, and ℓG(E0) = ∞ otherwise. We denote ℓG(E0) as ℓG(e) if E0 = {e}.
In this paper, we prove that if G has a spanning tree T such that ℓG(e) is odd for each e ∈ E(G) \E(T), the edges in E(G) \E(T) can be labeled as e1, e2, . . . , eq with ℓG(ei) ≤ ℓG(ei+1) for all 1 ≤ i ≤ q−1 and each edge ei is contained in a cycle Ci of size ℓG(ei) with E(Ci) ⊆ E(T) ∪ {ej : 1 ≤ j ≤ i}, then G is a graph in DP≈. As a direct application, all plane near-triangulations and complete multipartite graphs with at least three partite sets belong to DP≈. We also show that if E∗ is a set of edges in G such that ℓG(E∗) is even and E∗ satisfies certain conditions, then G belongs to DP<. In particular, if ℓG(E∗) = 4, where E∗ is a set of edges between two disjoint vertex subsets of G, then G belongs to DP<. Both results extend known ones by Dong and Yang.
In this paper, we prove that if G has a spanning tree T such that ℓG(e) is odd for each e ∈ E(G) \E(T), the edges in E(G) \E(T) can be labeled as e1, e2, . . . , eq with ℓG(ei) ≤ ℓG(ei+1) for all 1 ≤ i ≤ q−1 and each edge ei is contained in a cycle Ci of size ℓG(ei) with E(Ci) ⊆ E(T) ∪ {ej : 1 ≤ j ≤ i}, then G is a graph in DP≈. As a direct application, all plane near-triangulations and complete multipartite graphs with at least three partite sets belong to DP≈. We also show that if E∗ is a set of edges in G such that ℓG(E∗) is even and E∗ satisfies certain conditions, then G belongs to DP<. In particular, if ℓG(E∗) = 4, where E∗ is a set of edges between two disjoint vertex subsets of G, then G belongs to DP<. Both results extend known ones by Dong and Yang.
Date Issued
2023
Publisher
Wiley
Journal
Journal of Graph Theory
Funding Agency
Ministry of Education, Singapore