Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A said to an odd semi-clique it obtained from clique $2k+1$ vertices by deleting $k-1$ edges. Bonamy and Perrett asked every $\lfloor \frac{n}{2} \rfloor$ paths unless semi-clique. 2-degenerate subgraph has vertex de...

• Graph-based neural network models exploiting multiple self-supervised auxiliary tasks. We propose three new tasks for graph-based networks. Vertex features autoencoding. Corrupted vertex reconstruction. embeddings Self-supervised learning is currently gaining a lot of attention, as it allows networks to learn robust representations from large quantities unlabeled data. Additionally, multi-tas...

‎In this paper we defined the vertex removable cycle in respect of the following‎, ‎if $F$ is a class of graphs(digraphs)‎ ‎satisfying certain property‎, ‎$G in F $‎, ‎the cycle $C$ in $G$ is called vertex removable if $G-V(C)in in F $.‎ ‎The vertex removable cycles of eulerian graphs are studied‎. ‎We also characterize the edge removable cycles of regular‎ ‎graphs(digraphs).‎    

This is a continuation of previous study initiated by one us on nonlocal vertex bialgebras and smash product algebras. In this paper, we notion right $H$-comodule algebra for bialgebra $H$ give construction deformations algebras with structure compatible $H$-module structure. We also $\phi$-coordinated quasi modules As an example, family quantum deforming the associated to non-degenerate even l...

The notion of mixed representations of quivers can be derived from ordinary quiver representations by considering the dual action of groups on “vertex” vector spaces together with the usual action. A generating system for the algebra of semi-invariants of mixed representations of a quiver is determined. This is done by reducing the problem to the case of bipartite quivers of the special form an...

We study the automorphisms of a graph product of finitely-generated abelian groups W . More precisely, we study a natural subgroup Aut W of AutW , with Aut W = AutW whenever vertex groups are finite and in a number of other cases. We prove a number of structure results, including a semi-direct product decomposition Aut W = (InnW ⋊Out W )⋊Aut W . We also give a number of applications, some of wh...

The well-known Petersen G(5, 2) admits a semi-regular automorphism α acting on the vertex set with two orbits of equal size. This makes it a bicirculant. It is shown that trivalent bicirculants fall into four classes. Some basic properties of trivalent bicirculants are explored and the connection to combinatorial and geometric configurations are studied. Some analogues of the polycirculant conj...

We give proofs of Ore’s theorem on Hamilton circuits, Brooks’ theorem on vertex coloring, and Vizing’s theorem on edge coloring, as well as the Chvátal-Lovász theorem on semi-kernels, a theorem of Lu on spanning arborescences of tournaments, and a theorem of Gutin on diameters of orientations of graphs. These proofs, while not radically different from existing ones, are perhaps simpler and more...