# Complexity of frugal homomorphisms

## Date

2021-12-20

## Authors

Bard, Stefan

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## Abstract

Let t >= 1 be an integer. A k-colouring of a graph G is called t-frugal if no colour is assigned to more than t vertices in the neighbourhood of any vertex. Similarly, a homomorphism of G to H is t-frugal if no vertex of H is the image of more than t vertices in the neighbourhood of any vertex of G. This thesis is concerned with the complexity of frugal colourings and homomorphisms of graphs and digraphs.
We first consider t-frugal proper colourings. A colouring is proper if adjacent vertices must be assigned different colours. The proper t-frugal chromatic number of a graph is the minimum number of colours needed so that it has a proper t-frugal colouring. After determining the proper t-frugal chromatic number for trees or multipartite graphs, we present a polynomial time algorithm for computing the t-frugal chromatic number of a cograph. Next, we give a bound on the t-frugal chromatic number of any graph in terms of its maximum degree. When t is equal to the maximum degree of the graph, the given bound coincides with the bound arising from properly colouring the graph greedily. This bound is improved for K_4-minor-free graphs in a theorem which generalizes results of Lih, Wang and Zhu. Finally, a dichotomy theorem for the complexity of t-frugal k-colouring is proved in a theorem that generalizes the results of Fiala and Kratochvíl for the case t=1.
We next consider t-frugal colourings which are not proper colourings. Two variations are considered. In an improper open t-frugal k-colouring the frugality condition is applied with respect to the open neighbourhood of each vertex v. In an improper closed t-frugal k-colouring the frugality condition is applied with respect to the closed neighbourhood of each vertex v. Dichotomy theorems are proved for both variations. These theorems generalize results of Hahn, Kratochvíl, Širáň and Sotteau for the case t=1. The improper open t-frugal chromatic number of a graph is the minimum number of colours needed so that it has an improper open t-frugal colouring. The closed t-frugal chromatic number of a graph is defined similarly. Bounds for these numbers in terms of maximum degree are described. The improper closed 2-frugal chromatic number of cographs is determined, and a linear time algorithm for computing the improper closed 3-frugal chromatic number of a cograph is given. A linear time algorithm for computing the improper open 2-frugal chromatic number of a cograph is also given.
Our attention then shifts to t-frugal homomorphisms of connected graphs. The case when t=1 has been considered in the literature. No dichotomy theorem based on structural properties of the target graph is known. For t >= 2, we give a dichotomy theorem for the problem of deciding whether there exists a t-frugal homomorphism of a given graph G to a simple connected graph H. The problem is shown to be solvable in polynomial time if t = 2 and H is a tree of diameter at most 3, or if t >= 3 and the maximum degree of H is at most 1. It is shown to be NP-complete in all other cases.
Finally, we consider t-frugal homomorphisms of directed graphs. Here, there is a choice to be made about what t-frugal should mean. Following the paper of Courcelle and the thesis of Swarts, we consider the situation where the frugality condition is applied to in-neighbourhoods. For t >= 2 we prove a dichotomy theorem for the complexity of t-frugal homomorphism to a directed graph H which may contain loops. The problem of deciding whether a given directed graph G has a t-frugal homomorphism to a directed graph H is shown to be solvable in polynomial time if H has maximum in-degree at most 1, and NP-complete if H has maximum in-degree at least 2. This extends the dichotomy theorem of Swarts for the case where t=1 and every vertex of H has a loop. The thesis concludes by returning to improper t-frugal colourings, this time of directed graphs and in the case that the frugality condition is applied to in-neighbourhoods. For t >= 1, we describe a polynomial time algorithm to find the smallest number of colours needed so that a given tournament with a loop at each vertex has an improper t-frugal colouring.