Graph isomorphism problem is to determine whether two given graphs are isomorphic. It is a particular type of a more general problem “the isomorphism of incidence system”. I propose some new invariants for heuristic search of graph isomorphism and demonstrate that they are really useful in practice by experiments. Keyword. graph isomorphism, graph invariant, graph spectrum , adjacency matrix, m...

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. Let L(x) be a set of colors assigned to each element x of G. Then a list total coloring of G is a total coloring such that each element x receives a color contained in L(x). The list total coloring problem asks whether G has a list tot...

We study the weighted improper coloring problem, a generalization of defective coloring. We present some hardness results and in particular we show that weighted improper coloring is not fixed-parameter tractable when parameterized by pathwidth. We generalize bounds for defective coloring to weighted improper coloring and give a bound for weighted improper coloring in terms of the sum of edge w...

Graph coloring is the de facto standard technique for register allocation within a compiler. In this paper we examine the intuition that a better coloring algorithm results in better register allocation. By replacing the coloring phase of the gcc compiler’s register allocator with an optimal coloring algorithm, we demonstrate both the importance of extending the graph coloring model to better e...

The satisfiability problem is known to be NP-complete in general and for many restricted cases. One way to restrict instances of k-SAT is to limit the number of times a variable can occur. It was shown that for an instance of 4-SAT with the property that every literal appears in exactly 4 clauses (2 times negated and 2 times not negated), determining whether there is an assignment for literals,...

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of Grundy Coloring, the problem of determining whether a given graph has Grundy number at least k. We also study the variants Weak Grundy Coloring (where the coloring is not necessarily proper) and Connected Grundy Col...