Decomposition of (2k + 1)-regular graphs containing special spanning 2k-regular Cayley graphs into paths of length 2k + 1

On path decompositions of 2k-regular graphs

Tibor Gallai conjectured that the edge set of every connected graph G on n vertices can be partitioned into ⌈n/2⌉ paths. Let Gk be the class of all 2k-regular graphs of girth at least 2k − 2 that admit a pair of disjoint perfect matchings. In this work, we show that Gallai’s conjecture holds in Gk, for every k ≥ 3. Further, we prove that for every graph G in Gk on n vertices, there exists a par...

A Combinatorial Proof of a Relationship Between Maximal $(2k-1, 2k+1)$-Cores and $(2k-1, 2k, 2k+1)$-Cores

Integer partitions which are simultaneously t–cores for distinct values of t have attracted significant interest in recent years. When s and t are relatively prime, Olsson and Stanton have determined the size of the maximal (s, t)-core κs,t. When k > 2, a conjecture of Amdeberhan on the maximal (2k − 1, 2k, 2k + 1)-core κ2k−1,2k,2k+1 has also recently been verified by numerous authors. In this ...

On 2k-Twisted Graphs

We define the concept of a k-twisted chain in a (directed) graph and the concept of a 2k-twisted graph. We show that for a 2k-twisted graph the set of algebraic 2k-twisted cycles is an integral spanning set for the integral flow module of G. Since a graph is O-twisted if and only if it is strongly connected, our result generalizes the well-known theorem that there is a basis for the flow space ...

Efficient Construction of 2K+ Graphs

I. INTRODUCTION Researchers often need to generate synthetic graphs with key properties resembling those of networks of interest. This is for example the case when the full topology is unavailable or impractical to measure or use. In our work, we are interested in generating graphs that resemble online social networks in terms of their joint degree distribution (JDM) and their degree-dependent ...

Multiplier with Parallel CSA Using CRT's Specific Moduli (2k-1, 2k , 2k+1)