11 research outputs found

    Relating broadcast independence and independence

    Full text link
    An independent broadcast on a connected graph GG is a function f:V(G)→N0f:V(G)\to \mathbb{N}_0 such that, for every vertex xx of GG, the value f(x)f(x) is at most the eccentricity of xx in GG, and f(x)>0f(x)>0 implies that f(y)=0f(y)=0 for every vertex yy of GG within distance at most f(x)f(x) from xx. The broadcast independence number αb(G)\alpha_b(G) of GG is the largest weight ∑x∈V(G)f(x)\sum\limits_{x\in V(G)}f(x) of an independent broadcast ff on GG. Clearly, αb(G)\alpha_b(G) is at least the independence number α(G)\alpha(G) for every connected graph GG. Our main result implies αb(G)≤4α(G)\alpha_b(G)\leq 4\alpha(G). We prove a tight inequality and characterize all extremal graphs

    Eccentric Coloring in graphs

    Get PDF
    he \emph{eccentricity} e(u)e(u) of a vertex uu is the maximum distance of uu to any other vertex of GG. A vertex vv is an \emph{eccentric vertex} of vertex uu if the distance from uu to vv is equal to e(u)e(u). An \emph{eccentric coloring} of a graph G=(V,E)G = (V, E) is a function \emph{color}: V→N V \rightarrow N such that\\ (i) for all u,v∈Vu, v \in V, (color(u)=color(v))⇒d(u,v)>color(u)(color(u) = color(v)) \Rightarrow d(u, v) > color(u).\\ (ii) for all v∈Vv \in V, color(v)≤e(v)color(v) \leq e(v).\\ The \emph{eccentric chromatic number} χe∈N\chi_{e}\in N for a graph GG is the lowest number of colors for which it is possible to eccentrically color \ GG \ by colors: V→{1,2,…,χe}V \rightarrow \{1, 2, \ldots , \chi_{e} \}. In this paper, we have considered eccentric colorability of a graph in relation to other properties. Also, we have considered the eccentric colorability of lexicographic product of some special class of graphs
    corecore