Word-representability of face subdivisions of triangular grid graphs

A graph G = (V, E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if (x, y) ∈ E. A triangular grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A face subdivision of a triangular grid graph is replacing some of its cells by plane copies of the complete graph K4. Inspired by a recent elegant result of Akrobotu et al., who classified wordrepresentable triangulations of grid graphs related to convex polyominoes, we characterize word-representable face subdivisions of triangular grid graphs. A key role in the characterization is played by smart orientations introduced by us in this paper. As a corollary to our main result, we obtain that any face subdivision of boundary triangles in the Sierpi´nski gasket graph is wordrepresentable

Двухшаговая раскраска графов решетки различных типов

In this article, we consider the NP-hard problem of the two-step colouring of a graph. It is required to colour the graph in a given number of colours in a way, when no pair of vertices has the same colour, if these vertices are at a distance of 1 or 2 between each other. The optimum two-step colouring is one that uses the minimum possible number of colours.The two-step colouring problem is studied in application to grid graphs. We consider four types of grids: triangular, square, hexagonal, and octogonal. We show that the optimum two-step colouring of hexagonal and octogonal grid graphs requires 4 colours in the general case. We formulate the polynomial algorithms for such a colouring. A square grid graph with the maximum vertex degree equal to 3 requires 4 or 5 colours for a two-step colouring. In the paper, we suggest the backtracking algorithm for this case. Also, we present the algorithm, which works in linear time relative to the number of vertices, for the two-step colouring in 7 colours of a triangular grid graph and show that this colouring is always correct. If the maximum vertex degree equals 6, the solution is optimum.В данной статье рассматривается NP-трудная задача о двухшаговой раскраске графа. Она состоит в нахождении такой раскраски в заданное число цветов, при которой ни одна пара вершин на расстоянии 1 или 2 друг от друга не будет окрашена в одинаковый цвет. Оптимальной считается двухшаговая раскраска в минимально возможное количество цветов.Задача о двухшаговой раскраске исследуется применительно к графам решетки. Рассмотрены четыре типа решеток: треугольная, квадратная, шестиугольная и восьмиугольная. Показано, что в общем случае для оптимальной двухшаговой раскраски графов шестиугольной и восьмиугольной решетки требуется 4 цвета, приводятся полиномиальные алгоритмы получения такой раскраски. Для графа квадратной решетки, в котором максимальная степень вершины равна 3, может потребоваться 4 или 5 цветов для двухшаговой раскраски. В данной работе предложен алгоритм поиска с возвратом для этого случая. Для графов треугольной решетки представлен линейный относительно количества вершин алгоритм двухшаговой раскраски в 7 цветов, показано, что раскраска будет всегда корректной. Если максимальная степень вершины равна 6, решение будет оптимальным

Hamiltonian Paths in Some Classes of Grid Graphs

The Hamiltonian path problem for general grid graphs is known to be NP-complete. In this paper, we give necessary and sufficient conditions for the existence of Hamiltonian paths in L-alphabet, C-alphabet, F-alphabet, and E-alphabet grid graphs. We also present linear-time algorithms for finding Hamiltonian paths in these graphs

Hamiltonicity of locally hamiltonian and locally traceable graphs

Please read abstract in the article.The University of South Africa and the National Research Foundation of South Africa for their sponsorship of the Salt Rock Workshops of 28 July–10 August 2013 and 20–30 January 2016, which contributed towards results in this paper. The authors thank the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) for financial support, grant number BA2017/268. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the CoE-MaSS. This material is based upon the third author’s work supported by the National Research Foundation of S.A. under Grant number 81075 and the second author’s work supported by the National Research Foundation of S.A. under Grant number 107668.http://www.elsevier.com/locate/dam2019-02-19hj2018Mathematics and Applied Mathematic