Related papers: On Gy\'arf\'as' Path-Colour Problem

Let $G$ be a connected graph of chromatic number $k$. For a $k$-coloring $f$ of $G$, a full $f$-rainbow path is a path of order $k$ in $G$ whose vertices are all colored differently by $f$. We show that $G$ has a $k$-coloring $f$ such that…

Given a graph $G=(V,E)$ whose vertices have been properly coloured, we say that a path in $G$ is "colourful" if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy-Vitaver Theorem that every properly…

We prove that for all nonnegative integers k,s there exists c with the following property. Let G be a graph with clique number at most k and chromatic number more than c. Then for every vertex-colouring (not necessarily optimal) of G, some…

Answering a question raised by Dudek and Pra\l{}at, we show that if $pn\rightarrow \infty$, w.h.p.,~whenever $G=G(n,p)$ is $2$-coloured, there exists a monochromatic path of length $n(2/3+o(1))$. This result is optimal in the sense that…

Let G be an n-vertex graph that contains linearly many cherries (i.e., paths on 3 vertices), and let c be a coloring of the edges of the complete graph K_n such that at each vertex every color appears only constantly many times. In 1979,…

Let $G$ be an edge colored graph. A {\it}{rainbow path} in $G$ is a path in which all the edges are colored with distinct colors. Let $d^c(v)$ be the color degree of a vertex $v$ in $G$, i.e. the number of distinct colors present on the…

A graph $G$ arrows a graph $H$ if in every $2$-edge-coloring of $G$ there exists a monochromatic copy of $H$. Schelp had the idea that if the complete graph $K_n$ arrows a small graph $H$, then every "dense" subgraph of $K_n$ also arrows…

For any countably infinite graph $G$, Ramsey's theorem guarantees an infinite monochromatic copy of $G$ in any $r$-coloring of the edges of the countably infinite complete graph $K_\mathbb{N}$. Taking this a step further, it is natural to…

If $k\geq 0$, then a $k$-edge-coloring of a graph $G$ is an assignment of colors to edges of $G$ from the set of $k$ colors, so that adjacent edges receive different colors. A $k$-edge-colorable subgraph of $G$ is maximum if it is the…

Given a finite family $\mathcal{F}$ of graphs, we say that a graph $G$ is "$\mathcal{F}$-free" if $G$ does not contain any graph in $\mathcal{F}$ as a subgraph. A vertex-colored graph $H$ is called "rainbow" if no two vertices of $H$ have…

It is proved that every connected graph $G$ on $n$ vertices with $\chi(G) \geq 4$ has at most $k(k-1)^{n-3}(k-2)(k-3)$ $k$-colourings for every $k \geq 4$. Equality holds for some (and then for every) $k$ if and only if the graph is formed…

Let $G$ be an edge-colored graph. A rainbow (heterochromatic, or multicolored) path of $G$ is such a path in which no two edges have the same color. Let the color degree of a vertex $v$ be the number of different colors that are used on the…

The problem of finding the minimum number of colors to color a graph properly without containing any bicolored copy of a fixed family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of…

Resolving a problem raised by Norin, we show that for each $k \in \mathbb{N}$, there exists an $f(k) \le 7k$ such that every graph $G$ with chromatic number at least $f(k)+1$ contains a subgraph $H$ with both connectivity and chromatic…

Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no $(k-\epsilon)^{\operatorname{pw}(G)}\operatorname{poly}(n)$ time algorithm for deciding if an $n$-vertex graph $G$ with pathwidth $\operatorname{pw}(G)$ admits a proper vertex…

The Gy\'arf\'as-Sumner conjecture says that for every forest $H$ and every integer $k$, if $G$ is $H$-free and does not contain a clique on $k$ vertices then it has bounded chromatic number. (A graph is $H$-free if it does not contain an…

An equitable $k$-coloring of a graph is a proper $k$-coloring where the sizes of any two different color classes differ by at most one. In 1973, Meyer conjectured that every connected graph $G$ has an equitable $k$-coloring for some $k\leq…

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…

For a fixed integer, the $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for an integer $k$, such that no two adjacent vertices are coloured alike. A graph $G$ is $H$-free if $G$ does…

We investigate the extent to which the $k$-coloring graph $\mathcal{C}_{k}(G)$ uniquely determines the base graph $G$ and the number of colors $k$. The vertices of $\mathcal{C}_{k}(G)$ are the proper $k$-colorings of $G$, and edges connect…