156 research outputs found
Vertex-Magic Total Labeling on G-sun Graphs
Graph labeling is an immense area of research in mathematics, specifically graph theory. There are many types of graph labelings such as harmonious, magic, and lucky labelings. This paper will focus on magic labelings. Graph theorists are particularly interested in magic labelings because of a simple problem regarding tree graphs introduced in the 1990’s. The problem is still unsolved after almost thirty years. Researchers have studied magic labelings on other graphs in addition to tree graphs. In this paper we will consider vertex-magic labelings on G-sun graphs. We will give vertex-magic total labelings for ladder sun graphs and complete bipartite sun graphs. We will also show when there is no vertex-magic total labeling for other types of G-sun graphs
On the Graceful Game
A graceful labeling of a graph with edges consists of labeling the
vertices of with distinct integers from to such that, when each
edge is assigned as induced label the absolute difference of the labels of its
endpoints, all induced edge labels are distinct. Rosa established two well
known conjectures: all trees are graceful (1966) and all triangular cacti are
graceful (1988). In order to contribute to both conjectures we study graceful
labelings in the context of graph games. The Graceful game was introduced by
Tuza in 2017 as a two-players game on a connected graph in which the players
Alice and Bob take turns labeling the vertices with distinct integers from 0 to
. Alice's goal is to gracefully label the graph as Bob's goal is to prevent
it from happening. In this work, we study winning strategies for Alice and Bob
in complete graphs, paths, cycles, complete bipartite graphs, caterpillars,
prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths
Topological Additive Numbering of Directed Acyclic Graphs
We propose to study a problem that arises naturally from both Topological
Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as
Lucky Labeling). Let be a digraph and a labeling of its vertices with
positive integers; denote by the sum of labels over all neighbors of
each vertex . The labeling is called \emph{topological additive
numbering} if for each arc of the digraph. The problem
asks to find the minimum number for which has a topological additive
numbering with labels belonging to , denoted by
.
We characterize when a digraph has topological additive numberings, give a
lower bound for , and provide an integer programming formulation for
our problem, characterizing when its coefficient matrix is totally unimodular.
We also present some families for which can be computed in
polynomial time. Finally, we prove that this problem is \np-Hard even when its
input is restricted to planar bipartite digraphs
Trade-Offs in Distributed Interactive Proofs
The study of interactive proofs in the context of distributed network computing is a novel topic, recently introduced by Kol, Oshman, and Saxena [PODC 2018]. In the spirit of sequential interactive proofs theory, we study the power of distributed interactive proofs. This is achieved via a series of results establishing trade-offs between various parameters impacting the power of interactive proofs, including the number of interactions, the certificate size, the communication complexity, and the form of randomness used. Our results also connect distributed interactive proofs with the established field of distributed verification. In general, our results contribute to providing structure to the landscape of distributed interactive proofs
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