NP-completeness of local colorings of graphs
Introduction
In this paper, only finite, undirected and simple graphs are considered. For a graph , and are the sets of vertices and edges of G, respectively. By we denote the maximum degree of graph G. We denote by and the order and size of a graph G, respectively. A graph of order n is said to be an n-vertex graph. A k-coloring of G is a function such that for any two adjacent vertices . The notations and terminologies not mentioned here can be found in [1].
A local k-coloring of a graph G is a function such that for each , , there exist with at least the size of the subgraph induced by S. Note that a local coloring is in particular a proper usual vertex coloring. The local chromatic number of G is .
Local colorings were introduced by Chartrand et al. [2], [3] and further studied in [7], [10], [11], [12], [13]. A local coloring is a usual coloring with two additional conditions: any induced path of length 2 must contain two vertices with colors differing by at least 2 and any triangle contains two vertices with colors that differ by at least 3. If the latter condition is dropped, one speaks of the so-called semi-matching colorings which were studied in [7]. In the class of triangle-free graphs, local colorings and semi-matching colorings thus form the same concept.
Therefore, the local chromatic number of G is slightly more global than the chromatic number of G since the conditions on colors that can be assigned to the vertices of G depend on subgraphs of order 2 and 3 in G rather than only on subgraphs of order 2.
Note that the definitions of local k-coloring and local chromatic number in this paper are different from the ones introduced by Erdös, et al. [5].
Many combinatorial problems are NP-complete for general graphs, and are unlikely to be solvable in polynomial time. But if one of the problems does have a polynomial time algorithm, then they all do [4], [9]. Moreover, there exist a number of NP-complete problems remain NP-complete when their domains are substantially restricted [6]. By definition, the local coloring is a natural generalization of standard coloring which is NP-complete even for planar graphs [6]. Thus, it is interesting to determine whether the local coloring problem is NP-complete.
Graph coloring deals with the fundamental problem of partitioning a set of objects into classes, according to certain rules. Time table, sequencing, and scheduling problem, in their many forms, are basically of this nature [8].
In this paper, we show that determining whether a graph has a local k-coloring for fixed or is NP-complete, where . Moreover, determining whether a planar graph has a local 5-coloring is NP-complete even restricted to the maximum degree or 8.
Section snippets
NP-completeness of LOCAL COLORING
We consider the complexity of the LOCAL k-COLORING problem, which is defined as follows.
LOCAL k-COLORING:
Instance: A graph , positive integer k.
Question: Does G admit a local k-coloring?
VERTEX k-COLORING:
Instance: A graph , positive integer k.
Question: Does G admit a k-coloring?
It is well known that VERTEX k-COLORING problem is an NP-complete problem. Since the LOCAL k-COLORING problem is similar to the VERTEX k-COLORING problem, we may expect that the LOCAL k-COLORING problem is
NP-completeness of LOCAL 4-COLORING
In this section, we establish the NP-completeness of LOCAL 4-COLORING by a reduction from 3-SAT.
3-SAT:
Instance: A set of clauses, each having three variables.
Question: Can the variables be assigned value true or false such that each clause has at least one true variable?
An instance of 3-SAT is satisfiable if there is a truth assignment of its variables such that each clause has at least one true variable.
We first establish a useful lemma as follows.
Lemma 3.1 Let f be a local 4-coloring of G with the
Discussion
Combining Theorem 2.5, Theorem 3.4, we have known that LOCAL k-COLORING is NP-complete for any odd k with or . Moreover, if , by the definition of a local coloring, it is easy to see that any connected graph G with at least three vertices has no local k-coloring. Thus, the computation complexity of LOCAL k-COLORING for or is even remains open.
Recall that VERTEX 3-COLORING is NP-complete for planar graphs even restricted to the maximum degree four [6]. Analogously, we proved
Acknowledgements
This work is supported by the National Key Research and Development Program under grants 2016YFB0800700 and 2017YFB0802300, National Natural Science Foundation of China under grants 61672050, 61632002, 61572046, 11361008 and 61762047, Applied Basic Research (Key Project) of Sichuan Provinc under grant 2017JY0095.
References (13)
- et al.
Coloring graphs with locally few colors
Discrete Math.
(1986) A generalization of Kneser's conjecture
Discrete Math.
(2011)- et al.
A note on local coloring of graphs
Inf. Process. Lett.
(2015) - et al.
Local coloring of Kneser graphs
Discrete Math.
(2008) - et al.
Graph Theory
(2008) - et al.
Local colorings of graphs
Util. Math.
(2005)