On coloring parameters of triangle-free planar (n,m)(n,m)-graphs

An $(n,m)$-graph is a graph with $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to another $(n,m)$-graph $H$ is a vertex mapping that preserves the adjacencies along with their types and directions. The order of a smallest (with respect to the number of vertices) such $H$ is the $(n,m)$-chromatic number of $G$.Moreover, an $(n,m)$-relative clique $R$ of an $(n,m)$-graph $G$ is a vertex subset of $G$ for which no two distinct vertices of $R$ get identified under any homomorphism of $G$. The $(n,m)$-relative clique number of $G$, denoted by $\omega_{r(n,m)}(G)$, is the maximum $|R|$ such that $R$ is an $(n,m)$-relative clique of $G$. In practice, $(n,m)$-relative cliques are often used for establishing lower bounds of $(n,m)$-chromatic number of graph families. Generalizing an open problem posed by Sopena [Discrete Mathematics 2016] in his latest survey on oriented coloring, Chakroborty, Das, Nandi, Roy and Sen [Discrete Applied Mathematics 2022] conjectured that $\omega_{r(n,m)}(G) \leq 2 (2n+m)^2 + 2$ for any triangle-free planar $(n,m)$-graph $G$ and that this bound is tight for all $(n,m) \neq (0,1)$.In this article, we positively settle this conjecture by improving the previous upper bound of $\omega_{r(n,m)}(G) \leq 14 (2n+m)^2 + 2$ to $\omega_{r(n,m)}(G) \leq 2 (2n+m)^2 + 2$, and by finding examples of triangle-free planar graphs that achieve this bound. As a consequence of the tightness proof, we also establish a new lower bound of $2 (2n+m)^2 + 2$ for the $(n,m)$-chromatic number for the family of triangle-free planar graphs.Comment: 22 Pages, 5 figure

Scalable partitioning for parallel position based dynamics

We introduce a practical partitioning technique designed for parallelizing Position Based Dynamics, and exploiting the ubiquitous multi-core processors present in current commodity GPUs. The input is a set of particles whose dynamics is influenced by spatial constraints. In the initialization phase, we build a graph in which each node corresponds to a constraint and two constraints are connected by an edge if they influence at least one common particle. We introduce a novel greedy algorithm for inserting additional constraints (phantoms) in the graph such that the resulting topology is q-colourable, where ˆ qˆ ≥ 2 is an arbitrary number. We color the graph, and the constraints with the same color are assigned to the same partition. Then, the set of constraints belonging to each partition is solved in parallel during the animation phase. We demonstrate this by using our partitioning technique; the performance hit caused by the GPU kernel calls is significantly decreased, leaving unaffected the visual quality, robustness and speed of serial position based dynamics

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

Defensive Alliances in Signed Networks

The analysis of (social) networks and multi-agent systems is a central theme in Artificial Intelligence. Some line of research deals with finding groups of agents that could work together to achieve a certain goal. To this end, different notions of so-called clusters or communities have been introduced in the literature of graphs and networks. Among these, defensive alliance is a kind of quantitative group structure. However, all studies on the alliance so for have ignored one aspect that is central to the formation of alliances on a very intuitive level, assuming that the agents are preconditioned concerning their attitude towards other agents: they prefer to be in some group (alliance) together with the agents they like, so that they are happy to help each other towards their common aim, possibly then working against the agents outside of their group that they dislike. Signed networks were introduced in the psychology literature to model liking and disliking between agents, generalizing graphs in a natural way. Hence, we propose the novel notion of a defensive alliance in the context of signed networks. We then investigate several natural algorithmic questions related to this notion. These, and also combinatorial findings, connect our notion to that of correlation clustering, which is a well-established idea of finding groups of agents within a signed network. Also, we introduce a new structural parameter for signed graphs, signed neighborhood diversity snd, and exhibit a parameterized algorithm that finds a smallest defensive alliance in a signed graph