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An acyclic colouring of a graph is a proper colouring of the graph, for which every cycle uses at least three colours. In this paper, we describe a method for constructing all 4-connected acyclically 4-colourable planar triangulations that have exactly four odd-vertices, except the ones that contain no adjacent odd-vertices. Unlike previous operations, our method successfully establishes a connection with (acyclic) 4-colourings. Moreover, we discuss a special class of graphs, called diamond triangulations, and give a necessary and sufficient condition for a diamond triangulation to be acyclically 4-colourable. [ABSTRACT FROM AUTHOR]

A graphGis uniquelyk-colourable if the chromatic number ofGiskandGhas only onek-colouring up to permutation of the colours. Aksionov [On uniquely 3-colorable planar graphs, Discrete Math. 20 (1977), pp. 209–216] conjectured that every uniquely 3-colourable planar graph with at least four vertices has two adjacent triangles. However, in the same year, Melnikov and Steinberg [L.S. Mel'nikov and R. Steinberg,One counterexample for two conjectures on three coloring, Discrete Math. 20 (1977), pp. 203–206.] disproved the conjecture by constructing a counterexample. In this paper, we prove that if a uniquely 3-colourable planar graphGhas at most 4 triangles thenGhas two adjacent triangles. Furthermore, for any, we construct a uniquely 3-colourable planar graph withktriangles and without adjacent triangles. [ABSTRACT FROM PUBLISHER]

This paper proposes a quick exact algorithm for the graph coloring problem. The algorithm is based on implicit enumerating the coloring sequences of graph and converting the conflict edges constraints into carry numbers, which avoid exponential unnecessary graph coloring sequences testing. Only polynomial memory is needed in our algorithm, and symmetrical coloring solutions obtained by color permutations were removed. The algorithm performs reasonably well on a set of 61 benchmark graphs. [ABSTRACT FROM AUTHOR]

The concept of a metric dimension was proposed to model robot navigation where the places of navigating agents can change among nodes. The metric dimension m d (G) of a graph G is the smallest number k for which G contains a vertex set W, such that | W | = k and every pair of vertices of G possess different distances to at least one vertex in W. In this paper, we demonstrate that m d (H D N 1 (n)) = 4 for n ≥ 2 . This indicates that in these types of hex derived sensor networks, the least number of nodes needed for locating any other node is four. [ABSTRACT FROM AUTHOR]

Due to its wide application, deep reinforcement learning (DRL) has been extensively studied in the motion planning community in recent years. However, in the current DRL research, regardless of task completion, the state information of the agent will be reset afterward. This leads to a low sample utilization rate and hinders further explorations of the environment. Moreover, in the initial training stage, the agent has a weak learning ability in general, which affects the training efficiency in complex tasks. In this study, a new hierarchical reinforcement learning (HRL) framework dubbed hierarchical learning based on game playing with state relay (HGR) is proposed. In particular, we introduce an auxiliary penalty to regulate task difficulty, and one training mechanism, the state relay mechanism, is designed. The relay mechanism can make full use of the intermediate states of the agent and expand the environment exploration of low-level policy. Our algorithm can improve the sample utilization rate, reduce the sparse reward problem, and thereby enhance the training performance in complex environments. Simulation tests are carried out on two public experiment platforms, i.e., MazeBase and MuJoCo, to verify the effectiveness of the proposed method. The results show that HGR significantly benefits the reinforcement learning (RL) area.