相关论文: Graph coloring with no large monochromatic compone…

For $p\in \mathbb{N}$, a coloring $\lambda$ of the vertices of a graph $G$ is {\em{$p$-centered}} if for every connected subgraph~$H$ of $G$, either $H$ receives more than $p$ colors under $\lambda$ or there is a color that appears exactly…

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log…

We say a graph is $(d, d, \ldots, d, 0, \ldots, 0)$-colorable with $a$ of $d$'s and $b$ of $0$'s if $V(G)$ may be partitioned into $b$ independent sets $O_1,O_2,\ldots,O_b$ and $a$ sets $D_1, D_2,\ldots, D_a$ whose induced graphs have…

The \textit{square} of a graph $G$, denoted by $G^2$, is obtained from $G$ by adding an edge to connect every pair of vertices with a common neighbor in $G$. In this paper we prove that for every planar graph $G$ with maximum degree at most…

A total graph is an ordered triple $(V_0, V_1, E)$, where $V_0, V_1$ are the sets of empty and full vertices, respectively, $V_0 \cap V_1 = \emptyset$, and the set of edges $E$ is a subset of \(\binom{V_0 \cup V_1}{2}\) $(E\cap(V_0 \cup…

A coloring of a connected graph $G$ is a function $f$ mapping the vertex set of $G$ into the set of all integers. For any subgraph $H$ of $G$, we denote the sum of the values of $f$ on the vertices of $H$ as $f(H)$. If for any integer $k\in…

The conflict-free chromatic index of a graph $G$ is the minimum number of colours in an edge colouring of $G$ such that the neighbourhood of every edge contains a colour appearing exactly once. Its vertex analogue is the conflict-free…

A strong edge-coloring of a graph $G$ is a coloring of the edges such that every color class induces a matching in $G$. The strong chromatic index of a graph is the minimum number of colors needed in a strong edge-coloring of the graph. In…

A clique-coloring of a graph $G$ is a coloring of the vertices of $G$ so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, $\mathcal{H}(G)$, of a graph $G$ has $V(G)$ as its set of vertices and the maximal…

We consider unavoidable chromatic patterns in $2$-colorings of the edges of the complete graph. Several such problems are explored being a junction point between Ramsey theory, extremal graph theory (Tur\'an type problems), zero-sum Ramsey…

We consider $m$-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case $m = 2$ was first studied by Alon et al., who applied this framework to obtain surprisingly…

For a graph $G$, the \emph{equitable chromatic number} of $G$, denoted by $\chi_e(G)$, is the smallest integer $k$ such that $G$ admits a proper $k$-coloring whose color classes differ in size by at most one. We prove that for every…

Hadwiger's conjecture from 1943 states that for every integer $t\ge1$, every graph either can be $t$-colored or has a subgraph that can be contracted to the complete graph on $t+1$ vertices. As pointed out by Paul Seymour in his recent…

We propose the notion of a majority $k$-edge-coloring of a graph $G$, which is an edge-coloring of $G$ with $k$ colors such that, for every vertex $u$ of $G$, at most half the edges of $G$ incident with $u$ have the same color. We show the…

An edge colouring $c$ of a graph $G$ is called conflic-free if every non-isolated edge of $G$ has a uniquely coloured neighbour in its open edge neighbourhood. The least number of colours admitting such a colouring is denoted by $\chi'_{\rm…

A coloring of a graph is an assignment of colors to its vertices such that adjacent vertices have different colors. Two colorings are equivalent if they induce the same partition of the vertex set into color classes. Let $\mathcal{A}(G)$ be…

An odd coloring of a graph $G$ is a proper vertex coloring $\varphi$ with the property that for each non-isolated vertex $v\in V(G)$, there exists a color $c$ such that the cardinality of $\varphi^{-1}(c)\cap N(v)$ is odd. The concept of…

An \emph{interval $t$-coloring} of a graph $G$ is a proper edge-coloring with colors $1,\dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors. A graph $G$ is called \emph{interval…

A vertex colouring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Tucker conjectured that if every automorphism of a connected, locally finite graph moves infinitely many vertices, then there is…

We study the average number $\mathcal{A}(G)$ of colors in the non-equivalent colorings of a graph $G$. We show some general properties of this graph invariant and determine its value for some classes of graphs. We then conjecture several…