We present in this paper a graph theoretical model of gene assembly, where (segments of) genes are distributed over a set of circular molecules. This model is motivated by the process of gene assembly in ciliates, but it is more general. In this model a set of circular DNA molecules is represented by a bicoloured and labelled graph consisting of cyclic graphs, and the recombination takes place ...

Given an ordered partition Π = {P1, P2, ..., Pt} of the vertex set V of a connected graph G = (V,E), the partition representation of a vertex v ∈ V with respect to the partition Π is the vector r(v|Π) = (d(v, P1), d(v, P2), ..., d(v, Pt)), where d(v, Pi) represents the distance between the vertex v and the set Pi. A partition Π of V is a resolving partition of G if different vertices of G have ...

the first ($pi_1$) and the second $(pi_2$) multiplicative zagreb indices of a connected graph $g$, with vertex set $v(g)$ and edge set $e(g)$, are defined as $pi_1(g) = prod_{u in v(g)} {d_u}^2$ and $pi_2(g) = prod_{uv in e(g)} {d_u}d_{v}$, respectively, where ${d_u}$ denotes the degree of the vertex $u$. in this paper we present a simple approach to order these indices for connected graphs on ...

LetG = (V,E) be a graph with a fixed orientation and let A be an abelian group. Let F (G,A) denote the set of all functions from E(G) to A. The graph G is A-connected if for any function f̄ ∈ F (G,A), there exists an A-flow f such that f(e) 6= f̄(e) for any e ∈ E(G). The graph G is A-colorable if for any function f ∈ F (G,A), there exists a function c : V (G) → A such that for any arc e = (u, v) ...

We construct a connected graph H such that (1) χ(H) = ω; (2) Kω, the complete graph on ω points, is not a minor of H . Therefore Hadwiger’s conjecture does not hold for graphs with infinite coloring number. 1. Notation In this note we are only concerned with simple undirected graphs G = (V,E) where V is a set and E ⊆ P2(V ) where P2(V ) = { {x, y} : x, y ∈ V and x 6= y } . We also require that ...

As a generalization of connected domination in a graph G we consider domination by sets having at most k components. The order γ c (G) of such a smallest set we relate to γc(G), the order of a smallest connected dominating set. For a tree T we give bounds on γ c (T ) in terms of minimum valency and diameter. For trees the inequality γ c (T ) ≤ n− k − 1 is known to hold, we determine the class o...