For any simple graph G, let D(G) denote the degree set {degG(v) : v ∈ V (G)}. Let S be a finite, nonempty set of positive integers. In this paper, we first determine the families of graphs G which are unicyclic, bipartite satisfying D(G) = S, and further obtain the graphs of minimum orders in such families. More general, for a given pair (S, T) of finite, nonempty sets of positive integers of the same cardinality, it is shown that there exists a bipartite graph B(X, Y ) such that D(X) = S, D(Y )...
Given a graph with colored edges, a Hamiltonian cycle is
called alternating if its successive edges differ in color. The problem
of finding such a cycle, even for 2-edge-colored graphs, is trivially
NP-complete, while it is known to be polynomial for 2-edge-colored
complete graphs. In this paper we study the parallel complexity of
finding such a cycle, if any, in 2-edge-colored complete graphs. We give
a new characterization for such a graph admitting an alternating
Hamiltonian cycle which allows...
It is known that finding a perfect matching in a general graph
is
-equivalent to finding a perfect matching
in a 3-regular ( cubic) graph.
In this paper we extend this result to both, planar and bipartite cases.
In particular we prove that the construction
problem for perfect matchings in planar graphs
is as difficult as in the case of planar cubic graphs
like it is known to be the case for the famous Map Four-Coloring problem.
Moreover we prove that the existence and construction...
We study the problem of finding an alternating path having given endpoints and passing through a given set of vertices in edge-colored graphs (a path is alternating if any two consecutive edges are in different colors). In particular, we show that this problem in NP-complete for 2-edge-colored graphs. Then we give a polynomial characterization when we restrict ourselves to 2-edge-colored complete graphs. We also investigate on (s,t)-paths through fixed vertices, i.e. paths of length s+t such that...
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