1,419 research outputs found
Labeling of graphs, sumset of squares of units modulo n and resonance varieties of matroids
This thesis investigates problems in a number of different areas of graph theory and its applications in other areas of mathematics. Motivated by the 1-2-3-Conjecture, we consider the closed distinguishing number of a graph G, denoted by dis[G]. We provide new upper bounds for dis[G] by using the Combinatorial Nullstellensatz. We prove that it is NP-complete to decide for a given planar subcubic graph G, whether dis[G] = 2. We show that for each integer t there is a bipartite graph G such that dis[G] \u3e t. Then some polynomial time algorithms and NP-hardness results for the problem of partitioning the edges of a graph into regular and/or locally irregular subgraphs are presented. We then move on to consider Johnson graphs to find resonance varieties of some classes of sparse paving matroids. The last application we consider is in number theory, where we find the number of solutions of the equation x21 + _ _ _ + x2 k = c, where c 2 Zn, and xi are all units in the ring Zn. Our approach is combinatorial using spectral graph theory
Sigma Partitioning: Complexity and Random Graphs
A of a graph is a partition of the vertices
into sets such that for every two adjacent vertices and
there is an index such that and have different numbers of
neighbors in . The of a graph , denoted by
, is the minimum number such that has a sigma partitioning
. Also, a of a graph is a
function , such that for every two adjacent
vertices and of , ( means that and are adjacent). The of , denoted by , is the minimum number such
that has a lucky labeling . It was
conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is -complete to decide whether for a given 3-regular
graph . In this work, we prove this conjecture. Among other results, we give
an upper bound of five for the sigma number of a uniformly random graph
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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