Related papers: Multiple 3-Coloring, an Approach to 4-Coloring of …
It follows from the work of Tait and the Four-Color-Theorem that a planar cubic graph is 3-edge-colorable if and only if it contains no bridge. We consider the question of which planar graphs are subgraphs of planar cubic bridgeless graphs,…
Triangulation graph staining is sufficient for planar graph staining. This article will focus on triangulation and the nature of the color change channel of the staining tool. By construction, the four colors of the vertex are converted…
A decomposition of a graph $G$ is a family of subgraphs of $G$ whose edge sets form a partition of $E(G)$. In this paper, we prove that every triangle-free planar graph $G$ can be decomposed into a $2$-degenerate graph and a matching.…
Two cycles are {\em adjacent} if they have an edge in common. Suppose that $G$ is a planar graph, for any two adjacent cycles $C_{1}$ and $C_{2}$, we have $|C_{1}| + |C_{2}| \geq 11$, in particular, when $|C_{1}| = 5$, $|C_{2}| \geq 7$. We…
Proper graph coloring assigns different colors to adjacent vertices of the graph. Usually, the number of colors is fixed or as small as possible. Consider applications (e.g. variants of scheduling) where colors represent limited resources…
The concept of DP-coloring of a graph is a generalization of list coloring introduced by Dvo\v{r}\'{a}k and Postle in 2015. Multiple DP-coloring of graphs, as a generalization of multiple list coloring, was first studied by Bernshteyn,…
We show that every $n$-vertex planar graph is 3-colourable with monochromatic components of size $O(n^{4/9})$. The best previous bound was $O(n^{1/2})$ due to Linial, Matou\v{s}ek, Sheffet and Tardos [Combin. Probab. Comput., 2008].
We give a linear-time algorithm to decide 3-colorability of a triangle-free graph embedded in a fixed surface, and a quadratic-time algorithm to output a 3-coloring in the affirmative case. The algorithms also allow to prescribe the…
A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph G and a vertex…
A graph is $(c_1, c_2, ..., c_k)$-colorable if the vertex set can be partitioned into $k$ sets $V_1,V_2, ..., V_k$, such that for every $i: 1\leq i\leq k$ the subgraph $G[V_i]$ has maximum degree at most $c_i$. We show that every planar…
A $k$-subcoloring of a graph is a partition of the vertex set into at most $k$ cluster graphs, that is, graphs with no induced $P_3$. 2-subcoloring is known to be NP-complete for comparability graphs and three subclasses of planar graphs,…
We conclude an investigation of Abrishami, Esperet, Giocanti, Hamman, Knappe and M\"oller studying the existence of periodic colourings of locally finite graphs. A colouring of a graph $\Gamma$ is periodic if the resulting coloured graph…
The well-known Steinberg's conjecture asserts that any planar graph without 4- and 5-cycles is 3 colorable. In this note we have given a short algorithmic proof of this conjecture based on the spiral chains of planar graphs proposed in the…
In this work, we introduce DPG-coloring using the concepts of DP-coloring and variable degeneracy to modify the proofs on the following papers: (i) DP-3-coloring of planar graphs without $4$, $9$-cycles and cycles of two lengths from $\{6,…
We show that the edges of every 3-connected planar graph except $K_4$ can be colored with two colors in such a way that the graph has no color preserving automorphisms. Also, we characterize all graphs which have the property that their…
Recently, Borodin, Kostochka, and Yancey (On $1$-improper $2$-coloring of sparse graphs. Discrete Mathematics, 313(22), 2013) showed that the vertices of each planar graph of girth at least $7$ can be $2$-colored so that each color class…
In this paper, we study the achromatic and the pseudoachromatic numbers of planar and outerplanar graphs as well as planar graphs of girth 4 and graphs embedded on a surface. We give asymptotically tight results and lower bounds for maximal…
DP-coloring (also known as correspondence coloring) of a simple graph is a generalization of list coloring. It is known that planar graphs without 4-cycles adjacent to triangles are 4-choosable, and planar graphs without 4-cycles are…
We construct a moduli space of four colorings on planar cubic graphs. More precisely, we introduce the notion of weak Hamiltonian, a generalization of Hamiltonian cycles, and relate it to 4-colorings. Weak Hamiltonians have a form of…
In this paper, we prove that planar graphs without cycles of length 4, 6, 9 are 3-colorable.