Research on graph coloring has largely concentrated on various variants of the chromatic number (such as distance coloring,edge coloring...).Research on the traditional chromatic number has mostly focused on upper and lower bounds of the chromatic number , or on defining a graph class by its chromatic number and then studying its properties.The most successful example is perfect graphs.Other related directions include converting graphs into polynomials and studying the eigenvalues of adjacency matrices.For graph operations we mainly have Join,Cartesian Product,Tensor Product,Strong Product,Lexicographic Product.For the join of two graphs, the chromatic number is the sum of the chromatic numbers of the two original graphs, but the connections between graphs are usually not complete. This paper focuses on the impact of arbitrary connections on the chromatic number of graphs; we can say it is the``arbitrary connections'' of two graphs, especially complete graphs, and establishes some formal necessary and sufficient conditions regarding the chromatic number and the clique number. This article uses Hall's condition to prove that the chromatic number of arbitrary connection between two complete graphs is equal to the clique number, and establishes the relationship between arbitrary connection between complete graphs and quadratic forms.
Yuyi Wang (Thu,) studied this question.