11 research outputs found
Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree
The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for
every tree there exists a natural number such that the following
holds: If is a -edge-connected simple graph with size divisible by
the size of , then can be edge-decomposed into subgraphs isomorphic to
. So far this conjecture has only been verified for paths, stars, and a
family of bistars. We prove a weaker version of the Tree Decomposition
Conjecture, where we require the subgraphs in the decomposition to be
isomorphic to graphs that can be obtained from by vertex-identifications.
We call such a subgraph a homomorphic copy of . This implies the Tree
Decomposition Conjecture under the additional constraint that the girth of
is greater than the diameter of . As an application, we verify the Tree
Decomposition Conjecture for all trees of diameter at most 4.Comment: 18 page
A Creative Review on Coprime (Prime) Graphs
Coprime labelings and Coprime graphs have been of interest since 1980s and got popularized by the Entringer-Tout Tree Conjecture. Around the same time Newman's coprime mapping conjecture was settled by Pomerance and Selfridge. This result was further extended to integers in arithmetic progression. Since then coprime graphs were studied for various combinatorial properties. Here, coprimality of graphs for classes of graphs under the themes: Bipartite with special attention to Acyclicity, Eulerian and Regularity. Extremal graphs under non-coprimality and Eulerian properties are studied. Embeddings of coprime graphs in the general graphs, the maximum coprime graph and the Eulerian coprime graphs are studied as subgraphs and induced subgraphs. The purpose of this review is to assimilate the available works on coprime graphs. The results in the context of these themes are reviewed including embeddings and extremal problems
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Combinatorics
Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session