Related papers: Local convergence of random graph colorings

An odd $k$-edge-coloring of a graph $G$ is a (not necessarily proper) edge-coloring with at most $k$ colors such that each non-empty color class induces a graph in which every vertex is of odd degree; similarly, if more than one color per…

We study a variation of the graph colouring problem on random graphs of finite average connectivity. Given the number of colours, we aim to maximise the number of different colours at neighbouring vertices (i.e. one edge distance) of any…

Coloring unit-disk graphs efficiently is an important problem in the global and distributed setting, with applications in radio channel assignment problems when the communication relies on omni-directional antennas of the same power. In…

We give sufficient conditions under which a random graph with a specified degree sequence is symmetric or asymmetric. In the case of bounded degree sequences, our characterisation captures the phase transition of the symmetry of the random…

A random geometric graph $G_n$ is given by picking $n$ vertices in $\mathbb{R}^d$ independently under a common bounded probability distribution, with two vertices adjacent if and only if their $l^p$-distance is at most $r_n$. We investigate…

For graph classes $P_1,...,P_k$, Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph $G$ can be partitioned into subsets $V_1,...,V_k$ so that $V_j$ induces a graph in the class $P_j$…

Given a graphical degree sequence ${\bf d}=(d_1,\ldots, d_n)$, let $G(n, {\bf d})$ denote a uniformly random graph on vertex set $[n]$ where vertex $ i$ has degree $d_i$ for every $1\le i\le n$. We give upper and lower bounds on the joint…

We say a proper coloring of a graph is distance-$k$ fall if every vertex is within distance $k$ of at least one vertex of every color. We show that if $G$ is a connected graph of order at least $3$ that is $3$-colorable, thenit has a…

The $\!{}\bmod k$ chromatic index of a graph $G$ is the minimum number of colors needed to color the edges of $G$ in a way that the subgraph spanned by the edges of each color has all degrees congruent to $1\!\!\pmod k$. Recently, the…

A coloring is distinguishing (or symmetry breaking) if no non-identity automorphism preserves it. The distinguishing threshold of a graph $G$, denoted by $\theta(G)$, is the minimum number of colors $k$ so that every $k$-coloring of $G$ is…

Let $G$ be a graph and c a proper k-coloring of G, i.e. any two adjacent vertices u and v have different colors c(u) and c(v). A proper k-coloring is a b-coloring if there exists a vertex in every color class that contains all the colors in…

A 2-distance list k-coloring of a graph is a proper coloring of the vertices where each vertex has a list of at least k available colors and vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance…

It is shown that any graph with maximum degree $\Delta$ in which the average degree of the induced subgraph on the set of all neighbors of any vertex exceeds $\frac{6k^2}{6k^2 + 1}\Delta + k + 6$ is either $(\Delta - k)$-colorable or…

We study the properties of random graphs where for each vertex a {\it neighbourhood} has been previously defined. The probability of an edge joining two vertices depends on whether the vertices are neighbours or not, as happens in Small…

We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$, a randomised…

A vertex coloring of a graph $G$ is called distinguishing if no non-identity automorphisms of $G$ can preserve it. The distinguishing number of $G$, denoted by $D(G)$, is the minimum number of colors required for such a coloring, and the…

Consider a graph $G = (V, E)$ and, for each vertex $v \in V$, a subset $\Sigma(v)$ of neighbors of $v$. A $\Sigma$-coloring is a coloring of the elements of $V$ so that vertices appearing together in some $\Sigma(v)$ receive pairwise…

In this paper, a $k$-edge-coloring of $G$ is any mapping $c:E(G)\longrightarrow [k]$. The edge-coloring $c$ of $G$ naturally defines a vertex-coloring $\sigma_{c}: V(G) \to \mathbb{N}$, where $\sigma_{c}(v)=\sum_{u\in N_G(v)}c(vu)$ for…

List colouring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-colouring, we seek many in parallel. Our explorations have uncovered a…

For a simple graph G = (V, E) and a positive integer k greater than or equal to 2, a coloring of vertices of G using exactly k colors such that each vertex has an equal number of neighbors of each color is called neighborhood-balanced…