Parameterized Saga of First-Fit and Last-Fit Coloring
Akanksha Agrawal, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh,, Shaily Verma

TL;DR
This paper develops fixed-parameter tractable algorithms for specific graph coloring problems, resolving open questions about their computational complexity on general and restricted graph classes.
Contribution
It provides the first FPT algorithms for Partial Grundy Coloring on general graphs and for Grundy Coloring on $K_{i,j}$-free graphs, answering longstanding open problems.
Findings
FPT algorithm for Partial Grundy Coloring on general graphs.
FPT algorithm for Grundy Coloring on $K_{i,j}$-free graphs.
Introduction of new structural theorems and representative-family sets.
Abstract
The classic greedy coloring (first-fit) algorithm considers the vertices of an input graph in a given order and assigns the first available color to each vertex in . In the {\sc Grundy Coloring} problem, the task is to find an ordering of the vertices that will force the greedy algorithm to use as many colors as possible. In the {\sc Partial Grundy Coloring}, the task is also to color the graph using as many colors as possible. This time, however, we may select both the ordering in which the vertices are considered and which color to assign the vertex. The only constraint is that the color assigned to a vertex is a color previously used for another vertex if such a color is available. Whether {\sc Grundy Coloring} and {\sc Partial Grundy Coloring} admit fixed-parameter tractable (FPT) algorithms, algorithms with running time , where is the number of…
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