An on-line competitive algorithm for coloring bipartite graphs without long induced paths

The existence of an on-line competitive algorithm for coloring bipartite graphs remains a tantalizing open problem. So far there are only partial positive results for bipartite graphs with certain small forbidden graphs as induced subgraphs. We propose a new on-line competitive coloring algorithm for $P_9$-free bipartite graphs

Online graph coloring against a randomized adversary

Electronic version of an article published as Online graph coloring against a randomized adversary. "International journal of foundations of computer science", 1 Juny 2018, vol. 29, núm. 4, p. 551-569. DOI:10.1142/S0129054118410058 © 2018 copyright World Scientific Publishing Company. https://www.worldscientific.com/doi/abs/10.1142/S0129054118410058We consider an online model where an adversary constructs a set of 2s instances S instead of one single instance. The algorithm knows S and the adversary will choose one instance from S at random to present to the algorithm. We further focus on adversaries that construct sets of k-chromatic instances. In this setting, we provide upper and lower bounds on the competitive ratio for the online graph coloring problem as a function of the parameters in this model. Both bounds are linear in s and matching upper and lower bound are given for a specific set of algorithms that we call “minimalistic online algorithms”.Peer ReviewedPostprint (author's final draft

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

A new algorithm for on-line coloring bipartite graphs

We first show that for any bipartite graph $H$ with at most five vertices there exists an on-line competitive algorithm for the class of $H$-free bipartite graphs. We then analyze the performance of an on-line algorithm for coloring bipartite graphs on various subfamilies. The algorithm yields new upper bounds for the on-line chromatic number of bipartite graphs. We prove that the algorithm is on-line competitive for $P_7$-free bipartite graphs, i.e., that do not contain an induced path on seven vertices. The number of colors used by the on-line algorithm for $P_6$-free and $P_7$-free bipartite graphs is, respectively, bounded by roughly twice and roughly eight times the on-line chromatic number. In contrast, it is known that there exists no competitive on-line algorithm to color $P_6$-free (or $P_7$-free) bipartite graphs, i.e., for which the number of colors is bounded by any function depending only on the chromatic number