Graph Realizability and Factor Properties Based on Degree Sequence

A graph is a structure consisting of a set of vertices and edges. Graph construction has been a focus of research for a long time, and generating graphs has proven helpful in complex networks and artificial intelligence. A significant problem that has been a focus of research is whether a given sequence of integers is graphical. Havel and Hakimi stated necessary and sufficient conditions for a degree sequence to be graphic with different properties. In our work, we have proved the sufficiency of the requirements by generating algorithms and providing constructive proof. Given a degree sequence, one crucial problem is checking if there is a graph realization with k-factors. For the degree sequence with a realizable k-factor, we analyze an algorithm that produces the realization and its k-factor. We then generate degree sequences having no realizations with connected k-factors. We also state the conditions for a degree sequence to have connected k-factors. In our work, we have also studied the necessary and sufficient conditions for a sequence of integer pairs to be realized as directed graphs. We have proved the sufficiency of the conditions by providing algorithms as constructive proofs for the directed graphs

A sufficient condition for the existence of fractional (g,f,n)(g,f,n)-critical covered graphs

In data transmission networks, the availability of data transmission is equivalent to the existence of the fractional factor of the corresponding graph which is generated by the network. Research on the existence of fractional factors under specific network structures can help scientists design and construct networks with high data transmission rates. A graph $G$ is called a fractional $(g,f)$-covered graph if for any $e\in E(G)$, $G$ admits a fractional $(g,f)$-factor covering $e$. A graph $G$ is called a fractional $(g,f,n)$-critical covered graph if after removing any $n$ vertices of $G$, the resulting graph of $G$ is a fractional $(g,f)$-covered graph. In this paper, we verify that if a graph $G$ of order $p$ satisfies $p\geq\frac{(a+b-1)(a+b-2)+(a+d)n+1}{a+d}$, $\delta(G)\geq\frac{(b-d-1)p+(a+d)n+a+b+1}{a+b-1}$ and $\delta(G)>\frac{(b-d-2)p+2\alpha(G)+(a+d)n+1}{a+b-2}$, then $G$ is a fractional $(g,f,n)$-critical covered graph, where $g,f:V(G)\rightarrow Z^{+}$ be two functions such that $a\leq g(x)\leq f(x)-d\leq b-d$ for all $x\in V(G)$, which is a generalization of Zhou's previous result [S. Zhou, Some new sufficient conditions for graphs to have fractional $k$-factors, International Journal of Computer Mathematics 88(3)(2011)484--490].Comment: 1

Some conditions implying stability of graphs

A graph $X$ is said to be unstable if the direct product $X\times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is non-trivially unstable if it is unstable, connected, non-bipartite, and distinct vertices have distinct sets of neighbours. In this paper, we prove two sufficient conditions for stability of graphs in which every edge lies on a triangle, revising an incorrect claim of Surowski and filling in some gaps in the proof of another one. We also consider triangle-free graphs, and prove that there are no non-trivially unstable triangle-free graphs of diameter 2. An interesting construction of non-trivially unstable graphs is given and several open problems are posed.Comment: 13 page

Matchings, factors and cycles in graphs

A matching in a graph is a set of pairwise nonadjacent edges, a k-factor is a k-regular spanning subgraph, and a cycle is a closed path. This thesis has two parts. In Part I (by far the larger part) we study sufficient conditions for structures involving matchings, factors and cycles. The three main types of conditions involve: the minimum degree; the degree sum of pairs of nonadjacent vertices (Ore-type conditions); and the neighbourhoods of independent sets of vertices. We show that most of our theorems are best possible by giving appropriate extremal graphs. We study Ore-type conditions for a graph to have a Hamilton cycle or 2-factor containing a given matching or path-system, and for any matching and single vertex to be contained in a cycle. We give Ore-type and neighbourhood conditions for a matching L of l edges to be contained in a matching of k edges (l 2) containing a given set of edges. We also establish neighbourhood conditions for the existence of a cycle of length at least k. A list-edge-colouring of a graph is an assignment of a colour to each edge from its own list of colours. In Part II we study edge colourings of powers of cycles, and prove the List-Edge-Colouring Conjecture for squares of cycles of odd length