Cooperative coloring of some graph families

In a family G (1) , G (2) , ... , G (m) of graphs sharing the same vertex set V , a cooperative coloring involves selecting one independent set I( i )from G (i) for each i is an element of { 1, 2, ..., m } such that U (m)(i = 1) I (i) = V . For a graph class Q, let m (G) ( d ) denote the minimum m required to ensure that any graph family G (1 ), G (2) , ... , G (m) on the same vertex set, where G( i) is an element of Q and Delta( G( i )) <= d for each i is an element of { 1, 2, ... , m } , admits a cooperative coloring. For the graph classes T (trees) and W (wheels), we find that m (T) ( 3 ) = 4 and m( W) ( 4 ) = 5. Also, we prove that m B & lowast; ( d ) = O ( log (2 )d ) ) and m( L) ( d ) = O ( log d /log log d), where B (& lowast;) represents the class of graphs whose components loglog d are balanced complete bipartite graphs, and L represents the class of graphs whose components are generalized theta graphs. (c) 2024 Elsevier B.V. All rights reserved.