Abstract: A -colouring of a graph is a vertex colouring with at most colours such that every monochromatic connected component has maximum degree at most . Thus, a -colouring corresponds to a proper -colouring. It is well known that every planar graph admits -colourings, but there is no for which every planar graph admits -colourings or -colourings. Moreover, deciding whether a planar graph admits a -colouring is NP-complete. It is also known that every outerplanar graph admits and -colourings, although some outerplanar graphs do not admit -colourings, with triangles playing an important role. In this work, we prove that deciding the existence of -colourings of apart-plane graphs - in which every pair of triangles is disjoint - is also NP-complete. Moreover, we present a polynomial-time algorithm for deciding whether an outerpath graph - an outerplane graph for which the weak dual is isomorphic to a path - admits -colourings. This fully characterises the -colourability of outerpath graphs and provides new insights into the structural role of triangles in defective colouring problems.

Abstract: Given a graph , the pair is a neighbour-distinguishing -edge-labelling if such that, for every , and for every edge . The least for which it has been shown that every graph admits a neighbour-distinguishing -edge-labelling is three. The -Conjecture, proposed in 2004 by Karoński et al., states that every graph has a neighbour-distinguishing -edge-labelling. This conjecture has been recently proved by Keusch and published May 2024. In 2017, Luiz and Campos verified the -Conjecture for powers of paths and conjectured that a neighbour-distinguishing -edge-labelling could be built for powers of paths not isomorphic to complete graphs. In this work, we prove this conjecture.

Abstract: We investigate the -total labelling for complete equipartite graphs . Motivated by the conjecture of Havet and Yu, which states that every graph satisfies , we provide constructive labellings that support this conjecture for almost all cases of with and . The only exception is when is even and is odd, for which we establish a new upper bound of .

Abstract: The Socioenactive approach is a new field of study exploring novel dimensions of the design and development of computational systems. Socioenactive systems are computational systems that consider the human brain, body, and the environment surrounding it in the design of interactions. This promising research area brings new perspectives to the Human-Computer Interaction field, studying phenomena that occur in humans' brains, bodies, and their environments, considering cognition, sensorimotor processes, perception, and emotions. In socio-emotional interactions, the emotion contagion phenomenon emerges through bodily expressions, in particular, through neurophysiological expressions. Artificial Intelligence (AI) is being used to analyze emotions from human neurophysiological data. This study conducts a literature review concerning socienactive systems associated with AI and the Emotional Contagion Phenomenon. Considering the PRISMA (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) protocol, we guided our search for relevant studies. We identified studies from four different search libraries. We conducted a screening process based on the titles and abstracts of these investigations. After applying the exclusion criteria, we filtered and selected several studies. We analyzed the selected studies based on their main keywords and related concepts. We analyzed these studies considering their relevance for future studies.

Abstract: Let be a matching of a graph and let and be the sets of -saturated and -unsaturated vertices of , respectively. A vertex is unlocked if there exists a vertex . The matching is unlocked if every vertex in is unlocked. We say that a graph is -fully-extendable if every unlocked matching in of size can be extended to a perfect matching of . In this work, we study -fully-extendable graphs of order , characterizing the cases and . For the case , we provide additional necessary conditions, based on degree bounds, as well as sufficient conditions, based on classes of graphs.

Abstract: A matching in a graph is said to be extendable to a perfect matching if there exists a perfect matching of such that . In this work, we study the extendability of matchings under a neighbourhood condition: no unsaturated vertex has all of its neighbours -saturated. Vandenbussche and West showed that, in the hypercube , any matching of size at most is extendable to a perfect matching if and only if it satisfies this condition. We extend their result to the cartesian product , by proving that every matching of size at most is extendable to a perfect matching if and only if it does not saturate the neighbourhood of any unsaturated vertex. Furthermore, we demonstrate that this bound is tight by constructing a matching of size in that satisfies the neighbourhood condition but is not extendable to a perfect matching.

Abstract: A dominating set of a graph is a subset such that every vertex in either belongs to or is adjacent to some vertex in . The domination number is the minimum cardinality of a dominating set of . An independent dominating set of is a dominating set that is also independent and is the cardinality of minimum independent dominating set of . The computational complexity of these problems has led to extensive research focused on establishing bounds or exact values for these parameters in for graph families, especially cubic graphs. Additionally, determining the gap between these two parameters is a challenging problem. In this work, we introduce a family of cubic graphs, called Goldberg Graphs , which generalizes the well-known Goldberg Snarks and show that, for these graphs, .