Let G be a graph of order n, maximum degree ∆ and minimum degree δ. Let P (G,λ) be the chromatic polynomial of G. It is known that the multiplicity of zero ‘0’ of P (G,λ) is one if G is connected; and the multiplicity of zero ‘1’ of P (G,λ) is one if G is 2-connected. Is the multiplicity of zero ‘2’ of P (G,λ) at most one if G is 3-connected? In this paper, we first construct an infinite family...

A graph G is said to be quasi 4 -connected if G is 3-connected and for each cutset K ~ V( G) with IKI = 3, K is the neighbourhood of a vertex of degree three and G K has precisely two components. It is evident that such graphs need not be 4-connected and yet they exhibit many of the properties of 4-connected graphs. In this paper, we show that, given a set, N ~ E( G) of four independent edges, ...

Let G = (V,E) be a graph. A set D ⊆ V is a total outer-connected dominating set of G if D is dominating and G[V −D] is connected. The total outer-connected domination number of G, denoted γtc(G), is the smallest cardinality of a total outer-connected dominating set of G. It is known that if T is a tree of order n ≥ 2, then γtc(T ) ≥ 2n 3 . We will provide a constructive characterization for tre...

Let G=(V(G),E(G)) be a connected simple undirected graph with non empty vertex set V(G) and edge set E(G). For a positive integer k, by an edge irregular total k-labeling we mean a function f : V(G)UE(G) --> {1,2,...,k} such that for each two edges ab and cd, it follows that f(a)+f(ab)+f(b) is different from f(c)+f(cd)+f(d), i.e. every two edges have distinct weights. The minimum k for which G ...

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colors needed to color its edges, so that every pair of vertices is connected by at least one path in which no two edges are colored the same. Our main result is that rc(G) ≤ dn2 e for any 2-connected graph with at least three vertices. We conjecture that rc(G) ≤ n/κ + C for a κ-connected graph G of order n, w...

A k-container C(u, v) of a graph G is a set of k-disjoint paths joining u to v. A k-container C(u, v) of G is a k∗-container if it contains all the vertices of G. A graph G is k∗-connected if there exists a k∗-container between any two distinct vertices. Let (G) be the connectivity of G. A graph G is super connected if G is i∗-connected for all 1 i (G). A bipartite graph G is k∗-laceable if the...

Let $G$ be a group. The order graph of $G$ is the (undirected)graph $Gamma(G)$,those whose vertices are non-trivial subgroups of $G$ and two distinctvertices $H$ and $K$ are adjacent if and only if either$o(H)|o(K)$ or $o(K)|o(H)$. In this paper, we investigate theinterplay between the group-theoretic properties of $G$ and thegraph-theoretic properties of $Gamma(G)$. For a finite group$G$, we s...

A hub set in a graph G is a set U ⊆ V (G) such that any two vertices outside U are connected by a path whose internal vertices lie in U . We prove that h(G) ≤ hc(G) ≤ γc(G) ≤ h(G) + 1, where h(G), hc(G), and γc(G), respectively, are the minimum sizes of a hub set in G, a hub set inducing a connected subgraph, and a connected dominating set. Furthermore, all graphs with γc(G) > hc(G) ≥ 4 are obt...

The diameter diam(H) of a (directed) graph H is the maximum value computed over the lengths of all shortest (directed) paths in H . Given a bridgeless connected graph G, the oriented diameter OD(G) is given by OD(G) = min{diam(H) : H is an orientation of G}. In this paper, we show that OD(G) ≤ 2diam(G) +1 for every connected bridgeless interval graph G. Further, for every 2-connected proper int...

A graph G is said to be k-γt-critical if the total domination number γt(G) = k and γt(G + uv) < k for every uv / ∈ E(G). A k-γc-critical graph G is a graph with the connected domination number γc(G) = k and γc(G + uv) < k for every uv / ∈ E(G). Further, a k-tvc graph is a graph with γt(G) = k and γt(G− v) < k for all v ∈ V (G), where v is not a support vertex (i.e. all neighbors of v have degre...