This paper concerns two related enumeration problems on vertex labeled graphs. Given such a graph G, we investigate the number C(G) of connected subsets of the vertex set and the number P (G) of connected partitions of the vertex set. By connected we mean that the induced subgraphs are connected. The numbers C(G) and P (G) can be regarded as the (connected) graph analogs of the number of subset...

It is known that the removal of an edge from a graph G cannot decrease a domination number γ(G) and can increase it by at most one. Thus we can write that γ(G) ≤ γ(G − e) ≤ γ(G) + 1 when an arbitrary edge e is removed. Here we present similar inequalities for the weakly connected domination number γw and the connected domination number γc, i.e., we show that γw(G) ≤ γw(G− e) ≤ γw(G) + 1 and γc(...

Let lpt(G) be the minimum cardinality of a set of vertices that intersects all longest paths in a graph G. Let ω(G) be the size of a maximum clique in G, and tw(G) be the treewidth of G. We prove that lpt(G) ≤ max{1,ω(G)− 2} when G is a connected chordal graph; that lpt(G) = 1 when G is a connected bipartite permutation graph or a connected full substar graph; and that lpt(G) ≤ tw(G) for any co...

‎Let $f$ be a proper $k$-coloring of a connected graph $G$ and‎ ‎$Pi=(V_1,V_2,ldots,V_k)$ be an ordered partition of $V(G)$ into‎ ‎the resulting color classes‎. ‎For a vertex $v$ of $G$‎, ‎the color‎ ‎code of $v$ with respect to $Pi$ is defined to be the ordered‎ ‎$k$-tuple $c_{{}_Pi}(v)=(d(v,V_1),d(v,V_2),ldots,d(v,V_k))$‎, ‎where $d(v,V_i)=min{d(v,x):~xin V_i}‎, ‎1leq ileq k$‎. ‎If‎ ‎distinct...

a set $wsubseteq v(g)$ is called a resolving set for $g$, if for each two distinct vertices $u,vin v(g)$ there exists $win w$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. the minimum cardinality of a resolving set for $g$ is called the metric dimension of $g$, and denoted by $dim(g)$. in this paper, it is proved that in a connected graph $...

Tutte introduced the theory of nowhere zero flows and showed that a plane graph G has a face k-coloring if and only if G has a nowhere zero A-flow, for any Abelian group A with |A| ≥ k. In 1992, Jaeger et al. [9] extended nowhere zero flows to group connectivity of graphs: given an orientationD of a graph G, if for any b : V (G) → Awith  v∈V (G) b(v) = 0, there always exists a map f : E(G) →...

An edge-colored graph G, where adjacent edges may be colored the same, is rainbow connected if any two vertices of G are connected by a path whose edges have distinct colors. The rainbow connection number rc(G) of a connected graph G is the smallest number of colors that are needed in order to make G rainbow connected. In this paper, we give a sharp upper bound that rc(G) ≤ ⌈n2 ⌉ for any 2-conn...

A (δ, g)-cage is a regular graph of degree δ and girth g with the least possible number of vertices. It was proved by Fu, Huang and Rodger that every (3, g)-cage is 3-connected. Moreover, the same authors conjectured that all (δ, g)-cages are δ-connected for every δ ≥ 3. As a first step towards the proof of this conjecture, Jiang and Mubayi showed that every (δ, g)-cage with δ ≥ 3 is 3-connecte...

An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow connected. Krivelevich and Yuster have shown that a connected graph G with n vertices has rc(G) < 20n δ(G) [M. Krivelevich and...

Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and 19 A∗ = A−{0}. A graph G is A-connected if G has an orientation D(G) such that for every 20 function b : V (G) → A satisfying Pv∈V (G) b(v) = 0, there is a function f : E(G) → A∗ 21 such that for each vertex v ∈ V (G), the total amount of f values on the edges directed 22 out from v minus the total amount of f va...