相关论文: Planar digraphs of digirth five are 2-colorable
Given $\eta \in [0, 1]$, a colouring $C$ of $V(G)$ is an $\eta$-majority colouring if at most $\eta d^+(v)$ out-neighbours of $v$ have colour $C(v)$, for any $v \in V(G)$. We show that every digraph $G$ equipped with an assignment of lists…
For integers $a\ge 2b>0$, a \emph{circular $a/b$-flow} is a flow that takes values from $\{\pm b, \pm(b+1), \dots, \pm(a-b)\}$. The Planar Circular Flow Conjecture states that every $2k$-edge-connected planar graph admits a circular…
A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvo\v{r}\'{a}k and Postle. Essentially, DP-coloring assigns an arbitrary matching between lists of colors at adjacent vertices, as opposed to only…
A hypergraph is 2-intersecting if any two edges intersect in at least two vertices. Blais, Weinstein and Yoshida asked (as a first step to a more general problem) whether every 2-intersecting hypergraph has a vertex coloring with a constant…
A graph G is list (b:a)-colorable if for every assignment of lists of size b to vertices of G, there exists a choice of an a-element subset of the list at each vertex such that the subsets chosen at adjacent vertices are disjoint. We prove…
Thomassen proved that all planar graphs are $5$-choosable. \v{S}krekovski strengthened the result by showing that all $K_{5}$-minor-free graphs are $5$-choosable. Dvo\v{r}\'{a}k and Postle pointed out that all planar graphs are…
Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of sets such that $|L(v)|\ge 5$ for every $v\in V(G)$. By an $L$-coloring of a subgraph $J$ of $G$ we mean a (proper) coloring $\phi$ of $J$ such that…
A graph is apex if there is a vertex whose deletion makes the graph planar, and doublecross if it can be drawn in the plane with only two crossings, both incident with the infinite region in the natural sense. In 1966, Tutte conjectured…
A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most two receive distinct colors. It is known that every planar graph with maximum degree D has a strong edge coloring with at most 4D + 4 colors. We…
A coloring of the edges of a graph $G$ in which every $K_{1,2}$ is totally multicolored is known as a proper coloring and a coloring of the edges of $G$ in which every $K_{1,2}$ and every $K_{2,2}$ is totally multicolored is called a…
We prove that if one colors each point of the Euclidean plane with one of five colors, then there exist two points of the same color that are either distance $1$ or distance $2$ apart.
We show that every 2-degenerate graph with maximum degree $\Delta$ has a strong edge-coloring with at most $8\Delta-4$ colors.
The only open case of Vizing's conjecture that every planar graph with $\Delta\geq 6$ is a class 1 graph is $\Delta = 6$. We give a short proof of the following statement: there is no 6-critical plane graph $G$, such that every vertex of…
In this paper, we study the flexibility of two planar graph classes $\mathcal{H}_1$, $\mathcal{H}_2$, where $\mathcal{H}_1$, $\mathcal{H}_2$ denote the set of all hopper-free planar graphs and house-free planar graphs, respectively. Let $G$…
An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by Fiam{\v{c}}ik (1978) and Alon, Sudakov and Zaks (2001) states that every simple graph…
Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that $\frac32\Delta+1$ colors are sufficient…
For a number $\ell\geq 2$, let $\mathcal{G}_{\ell}$ denote the family of graphs which have girth $2\ell+1$ and have no odd hole with length greater than $2\ell+1$. Plummer and Zha conjectured that every 3-connected and internally…
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…
A plane graph is l-facially k-colourable if its vertices can be coloured with k colours such that any two distinct vertices on a facial segment of length at most l are coloured differently. We prove that every plane graph is 3-facially…
Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G has girth at least six and all lists have size at least three, then there exists an L-coloring respecting at…