Equitable list point arboricity of graphs

A graph $G$ is list point $k$-arborable if, whenever we are given a $k$-list assignment $L(v)$ of colors for each vertex $v\in V(G)$, we can choose a color $c(v)\in L(v)$ for each vertex $v$ so that each color class induces an acyclic subgraph of $G$, and is equitable list point $k$-arborable if $G$ is list point $k$-arborable and each color appears on at most $\lceil |V(G)|/k\rceil$ vertices of $G$. In this paper, we conjecture that every graph $G$ is equitable list point $k$-arborable for every $k\geq \lceil(\Delta(G)+1)/2\rceil$ and settle this for complete graphs, 2-degenerate graphs, 3-degenerate claw-free graphs with maximum degree at least 4, and planar graphs with maximum degree at least 8.

💡 Research Summary

The paper introduces the concept of equitable list point arboricity, a natural extension of the well‑studied list point arboricity. For a graph G, a list point k‑arborable coloring assigns to each vertex v a color from a prescribed list L(v) of size k such that each color class induces an acyclic subgraph. The equitable version further requires that no color is used on more than ⌈|V(G)|/k⌉ vertices. The authors formulate two conjectures: (1) the list point arboricity ρₗ(G) never exceeds ⌈(Δ(G)+1)/2⌉, and (2) for any integer k ≥ ⌈(Δ(G)+1)/2⌉, every graph G is equitable list point k‑arborable.

The central technical tool is Lemma 4, a deletion‑extension principle. If a set S={x₁,…,x_k} of distinct vertices can be removed so that the remaining graph G−S already admits an equitable list point k‑coloring, and if each vertex x_i has at most 2i−1 neighbors outside S, then the coloring can be extended to the whole graph by coloring the vertices of S one by one. This lemma reduces the problem to finding a suitable ordering of vertices that respects the neighbor‑bound condition.

Using this lemma, the authors verify the conjectures for several important graph families:

1. Complete graphs Kₙ – For k ≥ ⌈n/2⌉, a simple sequential coloring ensures each color appears on at most two vertices, yielding ρₗ(Kₙ)=⌈n/2⌉ and establishing equitable list point k‑arborability.

2. 2‑degenerate graphs – By repeatedly selecting a vertex of degree ≤2 (possible because the graph is 2‑degenerate) and constructing the set S with the low‑degree vertex as x₁ and a neighbor as x_k, the neighbor‑bound condition of Lemma 4 is satisfied. Induction on the number of vertices then proves the result for all k ≥ ⌈(Δ+1)/2⌉.

3. 3‑degenerate claw‑free graphs with Δ≥4 – The claw‑free property guarantees that any vertex of degree 3 has two adjacent neighbors, forming a triangle. The authors place the degree‑3 vertex, one neighbor, and another neighbor into positions x₁, x_{k‑1}, and x_k of S, respectively, and fill the remaining positions with vertices of degree ≤3. The neighbor bounds hold, and Lemma 4 yields equitable list point k‑arborability for k ≥ max{⌈(Δ+1)/2⌉, 3}.

4. Planar graphs with Δ≥8 – A discharging argument is employed. Initial charge c(v)=d(v)−4 is assigned to each vertex and c(f)=d(f)−4 to each face. Several redistribution rules (R1–R4) move charge from high‑degree vertices and large faces to low‑degree vertices and small faces. The authors prove that after discharging every element has non‑negative charge, which forces structural properties such as: every 3‑vertex is adjacent only to Δ‑vertices, 3‑faces are incident only with vertices of degree at least 5, and certain configurations of low‑degree vertices cannot occur. These structural lemmas guarantee the existence of a set S satisfying Lemma 4’s conditions, and thus every planar graph with Δ≥8 is equitable list point k‑arborable for k ≥ max{⌈(Δ+1)/2⌉, 5}.

The paper concludes that the two conjectures hold for the above families, and that the bounds are sharp for complete graphs. While the general conjecture remains open, the deletion‑extension framework combined with careful structural analysis (especially the discharging method for planar graphs) provides a powerful approach that may be extended to broader classes of graphs. The results also highlight the interplay between list coloring, arboricity, and equitable distribution, opening new directions for research in improper list colorings and their equitable variants.