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An edge-coloring of a graph $ G $ with colors $ 1, \ldots, t $ is an interval $ t $-coloring if all colors are used and the colors of edges incident to each vertex of $ G $ are distinct and form an interval of integers. A graph $ G $ is interval colorable if it has an interval $ t $-coloring for some positive integer $ t $. For an interval colorable graph $ G $, the least and the greatest values of $ t $ for which $ G $ has an interval $ t $-coloring are denoted by $ w(G) $ and $ W(G) $. Let $ G $ be a graph with vertex set $ V(G) = \{u_1, \ldots, u_{m}\} $, $ m\geq2 $, and let $ h_m = (H_i)_{i\in\{1, \ldots, m\}} $ be a sequence of vertex-disjoint with $ V(H_i) = \{x_1^{(i)}, \ldots, x_{n_i}^{(i)}\} $, $ n_i\geq 1 $. The generalized lexicographic products $ G[h_m] $ of $ G $ and $ h_m $ is a simple graph with vertex set $ \cup_{i = 1}^{m}V(H_i) $, in which $ x_p^{(i)} $ is adjacent to $ x_q^{(j)} $ if and only if either $ u_i = u_j $ and $ x_p^{(i)}x_q^{(i)}\in E(H_i) $ or $ u_iu_j\in E(G) $. In this paper, we obtain several sufficient conditions for the generalized lexicographic product $ G[h_m] $ to have interval colorings. We also present some sharp bounds on $ w(G[h_m]) $ and $ W(G[h_m]) $ of $ G[h_m] $.

An edge-coloring of a graph G with colors 1 , ... , t is an interval t -coloring if all colors are used and the colors of edges incident to each vertex of G are distinct and form an interval of integers. A graph G is interval colorable if it has an interval t -coloring for some positive integer t. For an interval colorable graph G , the least and the greatest values of t for which G has an interval t -coloring are denoted by w (G) and W (G). Let G be a graph with vertex set V (G) = { u 1 , ... , u m } , m ≥ 2 , and let h m = (H i) i ∈ { 1 , ... , m } be a sequence of vertex-disjoint with V (H i) = { x 1 (i) , ... , x n i (i) } , n i ≥ 1. The generalized lexicographic products G [ h m ] of G and h m is a simple graph with vertex set ∪ i = 1 m V (H i) , in which x p (i) is adjacent to x q (j) if and only if either u i = u j and x p (i) x q (i) ∈ E (H i) or u i u j ∈ E (G). In this paper, we obtain several sufficient conditions for the generalized lexicographic product G [ h m ] to have interval colorings. We also present some sharp bounds on w (G [ h m ]) and W (G [ h m ]) of G [ h m ]. [ABSTRACT FROM AUTHOR]

Let G(V, E) be a graph with order n with no component of order 2. An edge k-labeling α: E(G) →{1,2,…,k} is called a modular irregular k-labeling of graph G if the corresponding modular weight function wt_ α:V(G) → Z_n defined by wt_ α(x) =Ʃ_(xyϵE(G)) α(xy) is bijective. The value wt_α(x) is called the modular weight of vertex x. Minimum k such that G has a modular irregular k-labeling is called the modular irregularity strength of graph G. In this paper, we define a modular irregular labeling on C_n⊙mK_1. Furthermore, we determine the modular irregularity strength of C_n⊙mK_1. Keywords: corona product; cycle; empty graph; modular irregular labeling; modular irregularity strength. Abstrak Diberikan graf G(V, E) dengan orde n dengan tidak ada komponen yang berorde 2. Sebuah pelabelan-k sisi α: E(G) →{1,2,…,k} disebut pelabelan-k tak teratur modular pada graf G jika fungsi bobot modularnya wt_ α:V(G) → Z_n dengan wt_ α(x) =Ʃ_(xyϵE(G)) α(xy) merupakan fungsi bijektif. Nilai wt_α(x) disebut bobot modular dari simpul x. Minimum dari k sehingga G mempunyai pelabelan-k tak teratur modular disebut dengan kekuatan ketakteraturan modular dari graf G. Pada tulisan ini, didefinisikan pelabelan tak teratur modular pada C_n⊙mK_1. Lebih lanjut, ditentukan kekuatan ketakteraturan modular dari C_n⊙mK_1. Kata Kunci: hasil kali korona; lingkaran, graf kosong; pelabelan tak teratur modular; kekuatan ketakteraturan modular.