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Let G be a graph of order n and size m. A γ-labeling of G is a one-to-one function f:V(G) → 0,1,2,...,m that induces a labeling f’: E(G) → 1,2,...,m of the edges of G defined by f’(e) = |f(u)-f(v)| for each edge e = uv of G. The value of a γ-labeling f is . The maximum value of a γ-labeling of G is defined as
;
while the minimum value of a γ-labeling of G is
;
The values and are determined for double stars . We present characterizations of connected graphs G of order n for which or .
Let G be a connected graph and let c:V(G) → 1,2,...,k be a coloring of the vertices of G for some positive integer k (where adjacent vertices may be colored the same). The color code of a vertex v of G (with respect to c) is the ordered (k+1)-tuple code(v) = (a₀,a₁,...,aₖ) where a₀ is the color assigned to v and for 1 ≤ i ≤ k, is the number of vertices adjacent to v that are colored i. The coloring c is called recognizable if distinct vertices have distinct color codes and the recognition number...
A 2-stratified graph is a graph whose vertex set has been partitioned into two subsets, called the strata or color classes of . Two -stratified graphs and are isomorphic if there exists a color-preserving isomorphism from to . A -stratified graph is said to be homogeneously embedded in a -stratified graph if for every vertex of and every vertex of , where and are colored the same, there exists an induced -stratified subgraph of containing and a color-preserving...
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