On homomorphisms of oriented graphs with respect to the push operation

Article Type

Research Article

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Discrete Mathematics


An oriented graph G⃗ is a directed graph without directed cycles of length at most 2 having set of vertices V(G⃗) and set of arcs A(G⃗). To push a vertex of an oriented graph is to reverse the orientation of the arcs incident to that vertex. If G′⃗ can be obtained by pushing a set of vertices of G⃗, then we say G⃗ is in a push relation with G′⃗. A mapping f:V(G⃗)→V(H⃗) is a pushable homomorphism of G⃗ to H⃗ if there exists a G′⃗ which is in a push relation with G⃗ such that uv∈A(G′⃗) implies f(u)f(v)∈A(H⃗). Klostermeyer and MacGillivray (2004) introduced pushable homomorphism and defined the pushable chromatic number of an oriented graph G⃗ as the minimum cardinality of V(H⃗) such that G⃗ admits a pushable homomorphism to an oriented graph H. In this article, we further study the same topic and answer some of the questions asked in the above mentioned work, including studies of pushable chromatic numbers for the family of outerplanar graphs with girth restrictions, cactus, planar graphs and planar graphs with girth at least eight.

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Open Access, Bronze

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