Related papers: Three-coloring triangle-free graphs on surfaces I.…

Let $c$ be an edge-colouring of a graph $G$ such that for every vertex $v$ there are at least $d \ge 2$ different colours on edges incident to $v$. We prove that $G$ contains a properly coloured path of length 2d or a properly coloured…

An \emph{interval $t$-coloring} of a multigraph $G$ is a proper edge coloring with colors $1,\dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors. A \emph{cyclic interval $t$-coloring}…

We consider the 4-precoloring extension problem in \emph{planar near-Eulerian-triangulations}, i.e., plane graphs where all faces except possibly for the outer one have length three, all vertices not incident with the outer face have even…

Gallai's colouring theorem states that if the edges of a complete graph are 3-coloured, with each colour class forming a connected (spanning) subgraph, then there is a triangle that has all 3 colours. What happens for more colours: if we…

This is the second in a sequence of three papers in which we prove the following generalization of Thomassen's 5-choosability theorem: Let $G$ be a graph embedded on a surface of genus $g$. Then $G$ can be $L$-colored, where $L$ is a…

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…

it is shown that each triangle-free 1-planar graph with maximum degree $\Delta\geq7$ can be $\Delta$-colorable by Discharging Method.

We prove a decomposition theorem for the class of triangle-free graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. We prove that every graph of girth at least~5 in this class is…

There is a sufficiently large $N\in h\mathbb{N}$ such that the following holds. If $G$ is a tripartite graph with $N$ vertices in each vertex class such that every vertex is adjacent to at least $2N/3+2h-1$ vertices in each of the other…

Hu and Li investigate the signed graph version of Erd$\ddot{\mathrm{o}}$s problem: Is there a constant $c$ such that every signed planar graph without $k$-cycles, where $4\leq k\leq c$, is $3$-colorable and prove that each signed planar…

Several classical constructions illustrate the fact that the chromatic number of a graph can be arbitrarily large compared to its clique number. However, until very recently, no such construction was known for intersection graphs of…

A proper edge $t$-coloring of a graph is a coloring of its edges with colors $1,2,...,t$ such that all colors are used, and no two adjacent edges receive the same color. For any integer $n\geq 3$, all possible values of $t$ are found, for…

A graph $G$ is $(a,b)$-choosable if for any color list of size $a$ associated with each vertices, one can choose a subset of $b$ colors such that adjacent vertices are colored with disjoint color sets. This paper shows an equivalence…

It is known that DP-coloring is a generalization of a list coloring in simple graphs and many results in list coloring can be generalized in those of DP-coloring. In this work, we introduce a relaxed DP-coloring which is a generalization if…

A graph $G$ is $k$-vertex-critical if $\chi(G)=k$, but $\chi(G')<k$ for every proper induced subgraph $G'$ of $G$. For a family of graphs $\mathcal{F}$, $G$ is $\mathcal{F}$-free if no graph $F \in \mathcal{F}$ is an induced subgraph of…

Answering a question by Letzter and Snyder, we prove that for large enough $k$ any $n$-vertex graph $G$ with minimum degree at least $\frac{1}{2k-1}n$ and without odd cycles of length less than $2k+1$ is $3$-colourable. In fact, we prove a…

This paper investigates an extremely classic NP-complete problem: How to determine if a graph G, where each vertex has a degree of at most 4, can be 3-colorable(The research in this paper focuses on graphs G that satisfy the condition where…

We give an exact characterization of 3-colorability of triangle-free graphs drawn in the torus, in the form of 186 "templates" (graphs with certain faces filled by arbitrary quadrangulations) such that a graph from this class is not…

Let $C$ be a cycle and $f : V(C) \rightarrow \{c_1,c_2,\ldots,c_k\}$ a proper $k$-colouring of $C$ for some $k \ge 4$. We say the colouring $f$ is safe if for any planar graph $G$ in which $C$ is an induced cycle, there exists a proper…

Given a 3-colorable graph $X$, the 3-coloring complex $B(X)$ is the graph whose vertices are all the independent sets which occur as color classes in some 3-coloring of $X$. Two color classes $C,D \in V(B(X))$ are joined by an edge if $C$…