A graph G = (V,E) is said to be weakly four-connected if G is 4-edgeconnected and G− x is 2-edge-connected for every x ∈ V . We prove that every weakly four-connected Eulerian graph has a 2-connected Eulerian orientation. This verifies a special case of a conjecture of A. Frank.

A graph G is N2-locally connected if for every vertex v in G, the edges not incident with v but having at least one end adjacent to v in G induce a connected graph. In 1990, Ryjác̆ek conjectured that every 3-connected N2-locally connected claw-free graph is hamiltonian. This conjecture is proved in this note.

Proof: “⇒” Suppose D = (V,A) is a strongly connected graph obtained as an orientation of G = (V,E). Then, since ∀S ⊂ V , |δ D(S)| ≥ 1 and |δ − D(S)| ≥ 1, we have |δG(S)| ≥ 2. Therefore, G is 2-edge-connected. “⇐” Let G be a 2-edge-connected graph. Then G has an ear-decomposition. In other words, G is either a cycle C or G is obtained from a 2-edge-connected graph G by adding an ear P (a path) c...

An edge-colored graph G is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow edgeconnected. We prove that if G has n vertices and minimum degree δ then rc(G) < 20n/δ. This solves open problems from [5] and...

In this study, we investigate the further properties of quasi irresolute topological groups defined in [20]. We show that if a group homomorphism f between quasi irresolute topological groups is irresolute at $e_G$, then $f$ is irresolute on $G$. Later we prove that in a semi-connected quasi irresolute topological group $(G,*,tau )$, if $V$ is any symmetric semi-open neighborhood of $e_G$, then...

. Let G be a connected graph. A set S ⊆ V (G) is an outer-connected 2-resolving hop dominating of if and = or the subgraph ⟨V (G)\S⟩ induced by (G)\S connected. The domination number G, denoted γ^c2Rh(G) smallest cardinality G. This study aims to combine concept with sets graphs. main results generated in this include characterization join, corona, edge corona lexicographic product graphs, as w...

the tutte polynomial of a graph g, t(g, x,y) is a polynomial in two variables defined for every undirected graph contains information about how the graph is connected. in this paper a simple formula for computing tutte polynomial of a benzenoid chain is presented.

let $g$ be an $(n,m)$-graph. we say that $g$ has property $(ast)$if for every pair of its adjacent vertices $x$ and $y$, thereexists a vertex $z$, such that $z$ is not adjacentto either $x$ or $y$. if the graph $g$ has property $(ast)$, thenits complement $overline g$ is connected, has diameter 2, and itswiener index is equal to $binom{n}{2}+m$, i.e., the wiener indexis insensitive of any other...

Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and A∗=A−{0}. A graph G is A-connected if G has an orientation D(G) such that for every function b :V (G) →A satisfying ∑ v∈V (G)b(v )=0, there is a function f :E(G) →A∗ such that for each vertex v ∈V (G), the total amount of f values on the edges directed out from v minus the total amount of f values on the edges di...

the k-th semi total point graph of a graph g, , ‎is a graph‎ obtained from g by adding k vertices corresponding to each edge and‎ connecting them to the endpoints of edge considered‎. ‎in this paper‎, a formula for laplacian polynomial of in terms of‎ characteristic and laplacian polynomials of g is computed‎, ‎where is a connected regular graph‎.the kirchhoff index of is also computed‎.