@article{Jin2024, 
author = {Meiqin Jin and Ping Chen and Shuangliang Tian},
title = {Interval edge colorings of the generalized lexicographic product of some graphs},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {11},
pages = {30597-30611},
keywords = {generalized lexicographic product, interval edge coloring, path, empty graph, regular graph},
url = {https://www.sciopen.com/article/10.3934/math.20241477},
doi = {10.3934/math.20241477},
abstract = {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)={u1,…,um},  m≥2, and let  hm=(Hi)i∈{1,…,m} be a sequence of vertex-disjoint with  V(Hi)={x1(i),…,xni(i)},  ni≥1. The generalized lexicographic products  G[hm] of  G and  hm is a simple graph with vertex set  ∪i=1mV(Hi), in which  xp(i) is adjacent to  xq(j) if and only if either  ui=uj and  xp(i)xq(i)∈E(Hi) or  uiuj∈E(G). In this paper, we obtain several sufficient conditions for the generalized lexicographic product  G[hm] to have interval colorings. We also present some sharp bounds on  w(G[hm]) and  W(G[hm]) of  G[hm].}
}