## List-3-Coloring Ordered Graphs with a Forbidden Induced Subgraph

##### Abstract

Abstract. The List-3-Coloring Problem is to decide, given a graph G and a list L(v) ⊆
{1, 2, 3} of colors assigned to each vertex v of G, whether G admits a proper coloring ϕ with
ϕ(v) ∈ L(v) for every vertex v of G, and the 3-Coloring Problem is the List-3-Coloring
Problem on instances with L(v) = {1, 2, 3} for every vertex v of G. The List-3-Coloring
Problem is a classical NP-complete problem, and it is well-known that while restricted to
H-free graphs (meaning graphs with no induced subgraph isomorphic to a fixed graph H), it
remains NP-complete unless H is isomorphic to an induced subgraph of a path. However, the
current state of art is far from proving this to be sufficient for a polynomial time algorithm;
in fact, the complexity of the 3-Coloring Problem on P8-free graphs (where P8 denotes the
eight-vertex path) is unknown. Here we consider a variant of the List-3-Coloring Problem
called the Ordered Graph List-3-Coloring Problem, where the input is an ordered graph,
that is, a graph along with a linear order on its vertex set. For ordered graphs G and H, we
say G is H-free if H is not isomorphic to an induced subgraph of G with the isomorphism
preserving the linear order. We prove, assuming H to be an ordered graph, a nearly complete
dichotomy for the Ordered Graph List-3-Coloring Problem restricted to H-free ordered
graphs. In particular, we show that the problem can be solved in polynomial time if H has at
most one edge, and remains NP-complete if H has at least three edges. Moreover, in the case
where H has exactly two edges, we give a complete dichotomy when the two edges of H share
an end, and prove several NP-completeness results when the two edges of H do not share an
end, narrowing the open cases down to three very special types of two-edge ordered graphs.

##### Collections

##### Cite this version of the work

Sepehr Hajebi, Yanjia Li, Sophie Spirkl
(2024).
List-3-Coloring Ordered Graphs with a Forbidden Induced Subgraph. UWSpace.
http://hdl.handle.net/10012/20413

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