Monochromatic Clique Decompositions of Graphs

Area of Science:

• Graph Theory
• Combinatorics
• Discrete Mathematics

Background:

• Introduces monochromatic H-decompositions in k-edge-colored graphs.
• Defines the problem of partitioning edge sets into monochromatic subgraphs.
• Builds upon prior work on monochromatic Kr-decompositions.

Purpose of the Study:

• To determine the minimum number of elements in a monochromatic H-decomposition for a given graph G.
• To solve this problem for the specific case where H consists of cliques.
• To analyze the behavior of such decompositions for large graph orders (n).

Main Methods:

• Focuses on graph theory and combinatorial techniques.
• Extends existing results by Liu and Sousa.
• Analyzes edge-colored graphs and their partitions.

Main Results:

• Solves the monochromatic H-decomposition problem for clique graphs H.
• Establishes bounds for the decomposition size (phi_k(n,H)).
• The solution is valid for sufficiently large graph orders (n >= n0(H)).

Conclusions:

• Provides a complete solution for monochromatic clique-tuple decompositions in k-edge-colored graphs.
• Demonstrates the existence of such decompositions with a limited number of elements.
• Highlights the significance of graph order in decomposition properties.