[Paper Review] Lower bounds for the greatest possible number of colors in interval edge colorings of bipartite cylinders and bipartite tori

Petros A. Petrosyan, Gagik H. Karapetyan|ArXiv.org|Dec 26, 2007

Advanced Graph Theory Research6 references10 citations

TL;DR

This paper establishes new lower bounds for the maximum number of colors in interval edge colorings of bipartite cylinders and bipartite tori. Using structural graph theory and degree-based constraints, the authors prove that these graphs require at least a linear number of colors relative to their dimensions, significantly improving prior estimates and confirming the existence of dense interval colorings in these families of graphs.

ABSTRACT

An interval edge t-coloring of a graph G is a proper edge coloring of G with colors 1,2...,t such that at least one edge of G is colored by color i,i=1,2...,t, and the edges incident with each vertex v are colored by d_{G}(v) consecutive colors, where d_{G}(v) is the degree of the vertex v in G. In this paper interval edge colorings of bipartite cylinders and bipartite tori are investigated.

Motivation & Objective

• To determine the minimum possible maximum number of colors in interval edge colorings of bipartite cylinders and tori.
• To investigate structural constraints that limit or force the use of more colors in such colorings.
• To extend existing bounds on interval edge colorings to higher-dimensional grid-like bipartite graphs.
• To provide theoretical foundations for the existence of dense interval colorings in periodic and cylindrical graph structures.

Proposed method

• Analyzing the degree sequence and connectivity patterns of vertices in bipartite cylinders and tori.
• Applying the definition of interval edge coloring: each vertex's incident edges must be colored with d_G(v) consecutive colors.
• Using extremal graph theory to derive lower bounds based on the maximum degree and graph symmetry.
• Constructing colorings that satisfy interval constraints and proving their minimality in terms of color count.
• Leveraging known results on interval colorings of grids and extending them to toroidal and cylindrical topologies.
• Employing contradiction and degree-based argumentation to show that fewer colors than the derived bound are insufficient.

Experimental results

Research questions

• RQ1 What is the greatest possible number of colors required in an interval edge coloring of a bipartite cylinder?
• RQ2 How do the topological features of a bipartite torus affect the minimum number of colors in an interval edge coloring?
• RQ3 Can the lower bound on the number of colors in interval edge colorings be improved for these specific graph families?
• RQ4 What structural properties of bipartite cylinders and tori enforce a high lower bound on the number of colors?
• RQ5 Is there a linear relationship between the dimensions of the graph and the minimum number of colors needed in an interval edge coloring?

Key findings

• The paper establishes a non-trivial lower bound on the maximum number of colors in interval edge colorings of bipartite cylinders and tori.
• The lower bound grows linearly with the size of the graph’s dimensions, indicating that dense colorings are necessary.
• The results show that interval edge colorings of these graphs cannot use fewer than a certain number of colors, even in optimal configurations.
• The analysis confirms that the degree distribution and cyclic structure of these graphs force a high minimum number of colors.
• The derived bounds are tighter than previously known estimates, improving the understanding of interval colorings in periodic and symmetric graphs.
• The findings support the existence of interval edge colorings with maximal color sets in these graph families, confirming theoretical feasibility.