相关论文: A Note on Graphs of Dichromatic Number 2

Wang and Lih in 2002 conjectured that every planar graph without adjacent triangles is 4-choosable. In this paper, we prove that every planar graph without any 4-cycle adjacent to two triangles is DP-4-colorable, which improves the results…

A colouring of a graph is "nonrepetitive" if for every path of even order, the sequence of colours on the first half of the path is different from the sequence of colours on the second half. We show that planar graphs have nonrepetitive…

Every planar simple graph with n vertices has at least 2^(n/9) Z5-colorings.

A colouring of a digraph as defined by Erdos and Neumann-Lara in 1980 is a vertex-colouring such that no monochromatic directed cycles exist. The minimal number of colours required for such a colouring of a loopless digraph is defined to be…

A graph is $(d_1, \ldots, d_k)$-colorable if its vertex set can be partitioned into $k$ nonempty subsets so that the subgraph induced by the $i$th part has maximum degree at most $d_i$ for each $i\in\{1, \ldots, k\}$. It is known that for…

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.

This paper proves the following result: If $G$ is a planar graph and $L$ is a $4$-list assignment of $G$ such that $|L(x) \cap L(y)| \le 2$ for every edge $xy$, then $G$ is $L$-colourable. This answers a question asked by Kratochv\'{i}l,…

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…

A connected $k$-chromatic graph $G$ is double-critical if for all edges $uv$ of $G$ the graph $G - u - v$ is $(k-2)$-colourable. The only known double-critical $k$-chromatic graph is the complete $k$-graph $K_k$. The conjecture that there…

In 1965, Vizing proved that every planar graph $G$ with maximum degree $\Delta\geq 8$ is edge $\Delta$-colorable. It is also proved that every planar graph $G$ with maximum degree $\Delta=7$ is edge $\Delta$-colorable by Sanders and Zhao,…

A graph is $(d_1, ..., d_r)$-colorable if its vertex set can be partitioned into $r$ sets $V_1, ..., V_r$ so that the maximum degree of the graph induced by $V_i$ is at most $d_i$ for each $i\in \{1, ..., r\}$. For a given pair $(g, d_1)$,…

A proof that every outerplanar graph is \Delta+2 colorable. This is slightly stronger then an unpublished result of Wang Shudong, Ma Fangfang, Xu Jin, and Yan Lijun proving the same for 2-connected outerplanar graphs.

We show that every k-dichromatic vertex-critical digraph on at most 2k-2 vertices has a disconnected complement. This answers a question of Bang-Jensen et al., and generalises a classical theorem of Gallai on undirected vertex-critical…

The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph $G$ has as vertex set the set of all possible $k$-colourings of $G$ and two colourings are adjacent if they differ on the colour of exactly one vertex. Cereceda conjectured…

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…

An $r$-regular graph is an $r$-graph, if every odd set of vertices is connected to its complement by at least $r$ edges. We prove for $r \in \{4,5\}$, every projective planar $r$-graph with no Petersen-minor is $r$-edge colorable.

Petru\v{s}evski and \v{S}krekovski \cite{odd9} recently introduced the notion of an odd colouring of a graph: a proper vertex colouring of a graph $G$ is said to be \emph{odd} if for each non-isolated vertex $x \in V(G)$ there exists a…

Let $\Gamma$ be an Abelian group and let $G$ be a simple graph. We say that $G$ is $\Gamma$-colorable if for some fixed orientation of $G$ and every edge labeling $\ell:E(G)\rightarrow \Gamma$, there exists a vertex coloring $c$ by the…

DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvo\v{r}\'ak and Postle (2017). In this paper, we prove that every planar graph $G$ without $4$-cycles adjacent to $k$-cycles is…

DP-coloring is a relatively new coloring concept by Dvo\v{r}\'ak and Postle and was introduced as an extension of list-colorings of (undirected) graphs. It transforms the problem of finding a list-coloring of a given graph $G$ with a…