Characterizations and Clique Coloring of Edge Intersection Graphs on a Triangular Grid

被引:0
|
作者
de Luca, Vitor Tocci Ferreira [1 ]
Mazzoleni, Maria Pia [2 ]
Oliveira, Fabiano de Souza [1 ]
Szwarcfiter, Jayme Luiz [1 ,3 ]
机构
[1] Univ Estado Rio de Janeiro, Rio De Janeiro, Brazil
[2] Univ Nacl La Plata, La Plata, Argentina
[3] Univ Fed Rio de Janeiro, Rio de Janeiro, Brazil
来源
INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS | 2024年
关键词
Triangular grid; Intersection graphs; Paths on a grid; Single Bend Paths; Clique Coloring; SINGLE BEND PATHS;
D O I
10.1007/s13226-024-00698-x
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We introduce a new class of intersection graphs, the edge intersection graphs of paths on a triangular grid, called EPGt graphs. We show similarities and differences from this new class to the well-known class of EPG graphs. A turn of a path at a grid point is called a bend. An EPGt representation in which every path has at most k bends is called a Bk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {B}_k$$\end{document}-EPGt representation and the corresponding graphs are called Bk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {B}_k$$\end{document}-EPGt graphs. We provide examples of B2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {B}_{{2}}$$\end{document}-EPG graphs that are B1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {B}_{{1}}$$\end{document}-EPGt. We characterize the representation of cliques with three vertices and chordless 4-cycles in B1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {B}_{{1}}$$\end{document}-EPGt representations. We also prove that B1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {B}_{{1}}$$\end{document}-EPGt graphs have Strong Helly number 3. Furthermore, we prove that B1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {B}_{{1}}$$\end{document}-EPGt graphs are 7-clique colorable.
引用
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页数:21
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