A Lower Bound for Radio kk-chromatic Number of an Arbitrary Graph

Radio $k$-coloring is a variation of Hale's channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph $G$, subject to certain constraints involving the distance between the vertices. Specifically, for any simple connected graph $G$ with diameter $d$ and apositive integer $k$, $1\leq k \leq d$, a radio $k$-coloring of $G$ is an assignment $f$ of positive integers to the vertices of $G$ such that $|f(u)-f(v)|\geq 1+k-d(u, v)$, where $u$ and $v$ are any two distinct vertices of $G$ and $d(u, v)$ is the distance between $u$ and $v$.In this paper we give a lower bound for the radio $k$-chromatic number of an arbitrarygraph in terms of $k$, the total number of vertices $n$ and apositive integer $M$ such that $d(u,v)+d(v,w)+d(u,w)\leq M$ for all $u,v,w\in V(G)$. If $M$ is the triameter we get a better lower bound. We also find the triameter $M$ for several graphs, and show that the lower bound obtained for these graphs is sharp for the case $k=d$

Antipodal Labelings for Cycles

[[abstract]]Let G be a graph with diameter d. An antipodal labeling of G is a function f that assigns to each vertex a non-negative integer (label) such that for any two vertices u and v, vertical bar f(u) - f (v)vertical bar >= d - d(u, v), where d(u, v) is the distance between u and v. The span of an antipodal labeling f is max{f (u) - f(v) : u, v epsilon V (G)}. The antipodal number for G, denoted by an(G), is the minimum span of an antipodal labeling for C. Let C(n) denote the cycle on n vertices. Chartrand et al. [4] determined the value of an(C(n)) for n equivalent to 2 (mod 4). In this article we obtain the value of an(C(n)) for n equivalent to 1 (mod 4), confirming a conjecture in [4]. Moreover, we settle the case n equivalent to 3 (mod 4), and improve the known lower bound and give an upper bound for the case n equivalent to 0 (mod 4).[[note]]SC

Antipodal Labelings for Cycles

Let G be a graph with diameter d. An antipodal labeling of G is a function f that assigns to each vertex a non-negative integer (label) such that for any two vertices u and v, it is satisfied that |f(u) − f(v) | ≥ d − d(u,v), where d(u,v) is the distance between u and v. The span of an antipodal labeling f is max{f(u) − f(v) : u,v ∈ V (G)}. The antipodal number for G, denoted by an(G), is the minimum span of an antipodal labeling for G. Let Cn denote the cycle on n vertices. Chartrand, Erwin and Zhang [4] determined the value of an(Cn) for n ≡ 2 (mod 4). In this article we determine the value of an(Cn) for n ≡ 1 (mod 4), confirming a conjecture of Chartrand el al. Moreover, we settle the case n ≡ 3 (mod 4); and improve the known lower bound and give an upper bound for the case n ≡ 0 (mod 4).