Removable Edges in 4-Connected Graphs

Research on structural characterizations of graphs is a very popular topic in graph theory. The concepts of contractible edges and removable edges of graphs are powerful tools to study the structure of graphs and to prove properties of graphs by induction. \ud In 1998, Yin gave a convenient method to construct 4-connected graphs by using the existence of removable edges and contractible edges. He showed that a 4-connected graph can be obtained from a 2-cyclic graph by the following four operations: (i) adding edges, (ii) splitting vertices, (iii) adding vertices and removing edges, and (iv) extending vertices. Based on the above operations, we gave the following definition of removable edges in 4-connected graphs. Definition: Let $G$ be a 4-connected graph. For an edge $e$ of $G$, we perform the following operations on $G$: First, delete the edge $e$ from $G$, resulting in the graph $G-e$; Second, for each vertex $x$ of degree 3 in $G-e$, delete $x$ from $G-e$ and then completely connect the 3 neighbors of $x$ by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by $G \ominus e$. If $G\ominus e$ is still 4-connected, then the edge $e$ is called "removable"; otherwise, $e$ is called "unremovable". In the thesis we get the following results: (1) we study how many removable edges may exist in a cycle of a 4-connected graph, and we give examples to show that our results are in some sense the best possible. (2) We obtain results on removable edges in a longest cycle of a 4-connected graph. We also show that for a 4-connected graph $G$ of minimum degree at least 5 or girth at least 4, any edge of $G$ is removable or contractible. (3) We study the distribution of removable edges on a Hamilton cycle of a 4-connected graph, and show that our results cannot be improved in some sense. (4) We prove that every 4-connected graph of order at least six except 2-cyclic graph with order 6 has at least $(4|G|+16)/7$ removable edges. We also give a structural characterization of 4-connected graphs for which the lower bound is sharp. (5) We study how many removable edges there are in a spanning tree of a 4-connected graph and how many removable edges exist outside a cycle of a 4-connected graph. We also give examples to show that our results can not be improved in some sense

Edge Roman domination on graphs

An edge Roman dominating function of a graph $G$ is a function $f\colon E(G) \rightarrow \{0,1,2\}$ satisfying the condition that every edge $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e')=2$. The edge Roman domination number of $G$, denoted by $\gamma'_R(G)$, is the minimum weight $w(f) = \sum_{e\in E(G)} f(e)$ of an edge Roman dominating function $f$ of $G$. This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad stating that if $G$ is a graph of maximum degree $\Delta$ on $n$ vertices, then $\gamma_R'(G) \le \lceil \frac{\Delta}{\Delta+1} n \rceil$. While the counterexamples having the edge Roman domination numbers $\frac{2\Delta-2}{2\Delta-1} n$, we prove that $\frac{2\Delta-2}{2\Delta-1} n + \frac{2}{2\Delta-1}$ is an upper bound for connected graphs. Furthermore, we provide an upper bound for the edge Roman domination number of $k$-degenerate graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic graphs. In addition, we prove that the edge Roman domination numbers of planar graphs on $n$ vertices is at most $\frac{6}{7}n$, which confirms a conjecture of Akbari and Qajar. We also show an upper bound for graphs of girth at least five that is 2-cell embeddable in surfaces of small genus. Finally, we prove an upper bound for graphs that do not contain $K_{2,3}$ as a subdivision, which generalizes a result of Akbari and Qajar on outerplanar graphs

Flows on Bidirected Graphs

The study of nowhere-zero flows began with a key observation of Tutte that in planar graphs, nowhere-zero k-flows are dual to k-colourings (in the form of k-tensions). Tutte conjectured that every graph without a cut-edge has a nowhere-zero 5-flow. Seymour proved that every such graph has a nowhere-zero 6-flow. For a graph embedded in an orientable surface of higher genus, flows are not dual to colourings, but to local-tensions. By Seymour's theorem, every graph on an orientable surface without the obvious obstruction has a nowhere-zero 6-local-tension. Bouchet conjectured that the same should hold true on non-orientable surfaces. Equivalently, Bouchet conjectured that every bidirected graph with a nowhere-zero $\mathbb{Z}$-flow has a nowhere-zero 6-flow. Our main result establishes that every such graph has a nowhere-zero 12-flow.Comment: 24 pages, 2 figure