
On Domination Coloring in Graphs
A domination coloring of a graph G is a proper vertex coloring of G such...
read it

Explicit 3colorings for exponential graphs
Let H=(V,E) denote a simple, undirected graph. The 3coloring exponentia...
read it

Color Refinement, Homomorphisms, and Hypergraphs
Recent results show that the structural similarity of graphs can be char...
read it

Canonization for Bounded and Dihedral Color Classes in Choiceless Polynomial Time
In the quest for a logic capturing PTime the next natural classes of str...
read it

Maximizing Communication Throughput in Tree Networks
A widely studied problem in communication networks is that of finding th...
read it

Motifs, Coherent Configurations and Second Order Network Generation
In this paper we illuminate some algebraiccombinatorial structure under...
read it

Probabilistic Image Colorization
We develop a probabilistic technique for colorizing grayscale natural im...
read it
Identifiability of Graphs with Small Color Classes by the WeisfeilerLeman Algorithm
As it is well known, the isomorphism problem for vertexcolored graphs with color multiplicity at most 3 is solvable by the classical 2dimensional WeisfeilerLeman algorithm (2WL). On the other hand, the prominent CaiFürerImmerman construction shows that even the multidimensional version of the algorithm does not suffice for graphs with color multiplicity 4. We give an efficient decision procedure that, given a graph G of color multiplicity 4, recognizes whether or not G is identifiable by 2WL, that is, whether or not 2WL distinguishes G from any nonisomorphic graph. In fact, we solve the much more general problem of recognizing whether or not a given coherent configuration of maximum fiber size 4 is separable. This extends our recognition algorithm to graphs of color multiplicity 4 with directed and colored edges. Our decision procedure is based on an explicit description of the class of graphs with color multiplicity 4 that are not identifiable by 2WL. The CaiFürerImmerman graphs of color multiplicity 4 appear here as a natural subclass, which demonstrates that the CaiFürerImmerman construction is not ad hoc. Our classification reveals also other types of graphs that are hard for 2WL. One of them arises from patterns known as (n_3)configurations in incidence geometry.
READ FULL TEXT
Comments
There are no comments yet.