相关论文: Perfect 3-Colorings on 4-Regular Graph of Order 8

We prove that 3-Coloring remains NP-hard on 4- and 5-regular planar Hamiltonian graphs, strengthening the results of Dailey [Disc. Math.'80] and Fleischner and Sabidussi [J. Graph. Theor.'02]. Moreover, we prove that 3-Coloring remains…

Total colouring of 4-regular circulant graphs is an interesting but challenging topic, and has attracted much attention. However, it still remains an open question to determine the total chromatic numbers of $C_{n}(1, 3)$, a subclass of…

Deciding whether a planar graph (even of maximum degree $4$) is $3$-colorable is NP-complete. Determining subclasses of planar graphs being $3$-colorable has a long history, but since Gr\"{o}tzsch's result that triangle-free planar graphs…

A vertex colouring of some graph is called perfect if each vertex of colour $i$ has exactly $a_{ij}$ neighbours of colour $j$. Being perfect imposes several restrictions on the colour incidence matrix $(a_{ij})$. We list several (old and…

A vertex coloring of a given simple graph $G=(V,E)$ with $k$ colors ($k$-coloring) is a map from its vertex set to the set of integers $\{1,2,3,\dots, k\}$. A coloring is called perfect if the multiset of colors appearing on the neighbours…

In this paper, perfect k-orthogonal colourings of tensor graphs are studied. First, the problem of determining if a given graph has a perfect 2-orthogonal colouring is reformulated as a tensor subgraph problem. Then, it is shown that if two…

We show that any orientation of a graph with maximum degree three has an oriented 9-colouring, and that any orientation of a graph with maximum degree four has an oriented 69-colouring. These results improve the best known upper bounds of…

We give a linear-time algorithm to decide 3-colorability (and find a 3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.

Motivated by the theory of crystallizations, we consider an equivalence relation on the class of $3$-regular colored graphs and prove that up to this equivalence (a) there exists a unique contracted 3-regular colored graph if the number of…

A vertex colouring of some graph is called perfect if each vertex of colour $i$ has the same number $a_{ij}$ of neighbours of colour $j$. Here we determine all perfect colourings of the edge graphs of the hypercube in dimensions 4 and 5 by…

In this paper we enumerate the parameter matrices of all perfect 2-colorings of the generalized Petersen graphs GP(n;3), where n>=7. We also give some basic results for GP(n; k).

The 2-colorable perfect matching problem asks whether a graph can be colored with two colors so that each node has exactly one neighbor with the same color as itself. We prove that this problem is NP-complete, even when restricted to…

We define a perfect coloring of a graph $G$ as a proper coloring of $G$ such that every connected induced subgraph $H$ of $G$ uses exactly $\omega(H)$ many colors where $\omega(H)$ is the clique number of $H$. A graph is perfectly colorable…

We consider the problem of existence of perfect $2$-colorings (equitable $2$-partitions) of Hamming graphs with given parameters. We start with conditions on parameters of graphs and colorings that are necessary for their existence. Next we…

In this article, we use a unified approach to prove several classes of planar graphs are DP-$3$-colorable, which extend the corresponding results on $3$-choosability.

We prove that the two-colouring number of any planar graph is at most 8. This resolves a question of Kierstead et al. [SIAM J. Discrete Math.~23 (2009), 1548--1560]. The result is optimal.

Let G be a planar triangle-free graph and let C be a cycle in G of length at most 8. We characterize all situations where a 3-coloring of C does not extend to a proper 3-coloring of the whole graph.

In this paper, we consider the problem of a star coloring. In general case the problems in NP-complete. We establish the star chromatic number for splitting graph of complete and complete bipartite graphs, as well of paths and cycles. Our…

A planar graph can be embedded in a piecewise linear manifold, and the lattice on each linear piece can be colored with 3-coloring. If a planar graph can be colored with multiple 3-coloring, i.e. coloring the graph in pieces with different…

Graph colorings are becoming an increasingly useful family of mathematical models for a broad range of applications, such as time tabling and scheduling, frequency assignment, register allocation, computer security and so on. Graph proper…