ISSN 2590-9770 The Art of Discrete and Applied Mathematics 6 (2023) #P2.14 https://doi.org/10.26493/2590-9770.1523.2d7 (Also available at http://adam-journal.eu) Note on strong product graph dimension * Jaroslav Nešetˇril † , Aleš Pultr Department of Applied Mathematics and ITI, MFF, Charles University, Malostranské nám. 24, 11800 Praha 1, Czech Republic Received 10 January 2022, accepted 23 August 2022, published online 24 January 2023 Abstract In this paper we define a new dimension of graphs based on the strong product. Strong product can be viewed as a categorical product in a modified category. Unlike in the stan- dard case where the system of basic generators (“simplest objects”) is very transparent but necessarily infinite, we have here a single generator. Using just a single generator would lead to increasing complexity value even for structurally very trivial objects just because of the size. Thus, for more satisfactory results, it is advisable to use a transparent infinite system. Such one is proposed and several estimates for the resulting dimension are proved. Keywords: Graph, dimension, combinatorial complexity, product. Math. Subj. Class.: 05C15, 05C75, 05E99, 06A07, 18A30, 18B99 Introduction I Representation of complex graphs by means of operations (constructions) from simple graphs is a favorite and useful topic of graph theoty. In this note we concentrate on repre- sentations of graphs by means of products of graphs from a transparent class which we call generators. Sometimes a complex class can be generated by products of copies of a single graph. E.g., for every k there exists a graph A k such that any graph with the chromatic number at most k is an induced subgraph of a finite power of A k in the most usual categor- ical product ([16]). For bipartite graphs one can take A k = P 4 (this choice of generator has a category theory background – subdirect irreducibility – to be explained later). From the combinatorial point of view, such representation may be ineffective since we have to take a large power (e.g. the logarithm of the number of vertices). For example, when generating posets, it is more convenient to use products of linear orders instead of just linear orders of size 2. * Dedicated to Wilfried Imrich. † Corresponding author. E-mail addresses: nesetril@kam.mff.cuni.cz (Jaroslav Nešetˇril), pultr@kam.mff.cuni.cz (Aleš Pultr) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/