Sazdanovic and Yip (2018) defined a categorification of Stanley’s chromatic symmetric function called the homology, given by suitable family representations group. In this paper we prove that, as conjectured Chandler, Sazdanovic, Stella (2019), if graph $G$ is non-planar, then its homology in bidegree (1,0) contains $\mathbb{Z}_2$-torsion. Our proof follows recursive argument based on Ku...

The problem of routing in graphs using node-disjoint paths has received a lot of attention and a polylogarithmic approximation algorithm with constant congestion is known for undirected graphs [Chuzhoy and Li 2016] and [Chekuri and Ene 2013]. However, the problem is hard to approximate within polynomial factors on directed graphs, for any constant congestion [Chuzhoy, Kim and Li 2016]. Recently...

We have been investigating the cryptographical properties of infinite families of simple graphs of large girth with the special colouring of vertices during the last 10 years. Such families can be used for the development of cryptographical algorithms (on symmetric or public key modes) and turbocodes in error correction theory. Only few families of simple graphs of large unbounded girth and arb...

In 2001, Davies, Gladwell, Leydold, and Stadler proved discrete nodal domain theorems for eigenfunctions of generalized Laplacians, i.e., symmetric matrices with non-positive off-diagonal entries. this paper, we establish arbitrary by exploring the induced signed graph structure. Our concepts domains any function on a are switching invariant. When is balanced, our definitions upper bound estima...