Related papers: On uniquely list colorable graphs

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For $\alpha \leq 1$ and $k \in \mathbb{Z}^+$, we say that a graph $G=(V,E)$ is…

A path in an edge-colored graph $G$ is called monochromatic if any two edges on the path have the same color. For $k\geq 2$, an edge-colored graph $G$ is said to be monochromatic $k$-edge-connected if every two distinct vertices of $G$ are…

In this note, we revisit the notion of strong set-colorings introduced by Hegde (2009) and completed by equivalences due to Boutin et al. (2010) and provide a necessary and sufficient \emph{Steiner packing} characterisation: a finite graph…

The set of semialgebraic graphs having countable list-chromatic numbers is characterized. Some other related sets of graphs having countable list-chromatic numbers also are.

Let G be a plane graph with outer cycle C, let u,v be vertices of C and let (L(x):x in V(G)) be a family of sets such that |L(u)|=|L(v)|=2, L(x) has at least three elements for every vertex x of C-{u,v} and L(x) has at least five elements…

Let ${\mathcal D}_d$ be the class of $d$-degenerate graphs and let $L$ be a list assignment for a graph $G$. A colouring of $G$ such that every vertex receives a colour from its list and the subgraph induced by vertices coloured with one…

We prove for k at most 10, that every graph of chromatic number k with a unique k-coloring admits a clique minor of order k.

For a flexible labeling of a graph, it is possible to construct infinitely many non-equivalent realizations keeping the distances of connected points constant. We give a combinatorial characterization of graphs that have flexible labelings.…

We classify the countable ultrahomogeneous 2-vertex-colored graphs in which the color classes are imprimitive, i.e., up to complementation they form disjoint unions of cliques. This generalizes work by Jenkinson, Lockett and Truss as well…

A graph is k-total colourable if there is an assignment of k different colours to the vertices and edges of the graph such that no two adjacent nor incident elements receive the same colour. The total chromatic number of some direct product…

A k-fold x-coloring of a graph is an assignment of (at least) k distinct colors from the set {1, 2, ..., x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest number x such that G admits a…

A list assignment $L$ for a graph $G$ is an $(\ell,k)$-list assignment if $|L(v)|\geq \ell$ for each $v \in V(G)$ and $|L(u) \cap L(v)| \leq k$ for each $uv \in E(G)$. We say $G$ is $(\ell,k)$-choosable if it admits an $L$-colouring for…

A matching $M$ in a graph $G$ is uniquely restricted if there is no matching $M'$ in $G$ that is distinct from $M$ but covers the same vertices as $M$. Solving a problem posed by Golumbic, Hirst, and Lewenstein, we characterize the graphs…

Given a graph $G$ and a list assignment $L(v)$ for each vertex of $v$ of $G$. A proper $L$-list-coloring of $G$ is a function that maps every vertex to a color in $L(v)$ such that no pair of adjacent vertices have the same color. We say…

We study a new variant of graph coloring by adding a connectivity constraint. A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph $G$ is said to be…

Proportional choosability is a list coloring analogue of equitable coloring. Specifically, a $k$-assignment $L$ for a graph $G$ specifies a list $L(v)$ of $k$ available colors to each $v \in V(G)$. An $L$-coloring assigns a color to each…

Let $r \geqslant 0$ and $k \geqslant 1$ be integers. We say that a graph $G$ has an $r$-equitable $k$-coloring if there exists a proper $k$-coloring of $G$ such that the sizes of any two color classes differ by at most $r$. The least $k$…

For $k\in\mathbb{Z}^+$ and $G$ a simple connected graph, a $k$-radio labeling $f:V_G\to\Z^+$ of $G$ requires all pairs of distinct vertices $u$ and $v$ to satisfy $|f(u)-f(v)|\geq k+1-d(u,v)$. When $k=1$, this requirement gives rise to the…

A coloring of a direct product of graphs is said to be {\em trivial} iff it is induced by some coloring of a factor of the product. A graph $G$ is trivially power colorable iff every coloring of a finite power of $G$ with $\chi(G)$-many…

The Unfriendly Partition Conjecture posits that every countable graph admits a 2-colouring in which for each vertex there are at least as many bichromatic edges containing that vertex as monochromatic ones. This is not known in general, but…