相关论文: Geochromatic Number when Crossings are Independent

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of a graph $G$, is the smallest integer $k$ for which…

A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $\chi_p(G)$, of…

The boxicity (respectively cubicity) of a graph $G$ is the minimum non-negative integer $k$, such that $G$ can be represented as an intersection graph of axis-parallel $k$-dimensional boxes (respectively $k$-dimensional unit cubes) and is…

For an arbitrary invariant $\rho (G)$ of a graph $G$, the $\rho-$vertex stability number $vs_{\rho}(G)$ is the minimum number of vertices of $G$ whose removal results in a graph $H\subseteq G$ with $\rho (H)\neq \rho (G)$ or with…

For graphs $G$ and $H$, a homomorphism from $G$ to $H$, or $H$-coloring of $G$, is a map from the vertices of $G$ to the vertices of $H$ that preserves adjacency. When $H$ is composed of an edge with one looped endvertex, an $H$-coloring of…

The crossing number of a graph $G$, ${\mbox{cr}}(G)$, is the minimum number of crossings, the pair-crossing number, ${\mbox{pcr}}(G)$, is the minimum number of pairs of crossing edges over all drawings of $G$. In this note we show that…

The strong chromatic index of a graph $G$, denoted $\chi_s'(G)$, is the least number of colors needed to edge-color $G$ so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted…

The problem of chromatic thresholds seeks for minimum degree conditions that ensure $H$-free graphs to have a bounded chromatic number, or equivalently a bounded size homomorphic image. The strengthened homomorphism thresholds problem…

A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $\chi_p(G)$, of…

An $(n,m)$-graph is a graph with $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to another $(n,m)$-graph $H$ is a vertex mapping that preserves the adjacencies along with their types and directions. The…

Let $f$ be a proper $k$-coloring of a connected graph $G$ and $\Pi=(V_1,V_2,...,V_k)$ be an ordered partition of $V(G)$ into the resulting color classes. For a vertex $v$ of $G$, the color code of $v$ with respect to $\Pi$ is defined to be…

An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident with distinct sets of colors. The minimum number of colors needed for an adjacent vertex…

A vertex coloring of a graph is said to be \textit{conflict-free} with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al.,…

A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. It is known that the list edge chromatic number $\chi'_l(G)$ of any outer-1-planar graph $G$ with…

The chromatic discrepancy of a graph $G$, denoted $\phi(G)$, is the least over all proper colourings $\sigma$ of $G$ of the greatest difference between the number of colours $|\sigma(V(H))|$ spanned by an induced subgraph $H$ of $G$ and its…

The crossing number is the smallest number of pairwise edge-crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant…

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ without bichromatic paths or cycles of length four. The it star chromatic index, $\chi_{st}^{'} (G ),$ of $G$ is the minimum number $k$ for which $G$ has a star edge…

A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number. Given $n$ points $x_1,…

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of $G$, is the smallest integer $k$ for which $G$…

The interplay of minimum degree conditions and structural properties of large graphs with forbidden subgraphs is a central topic in extremal graph theory. For a given graph $F$ we define the homomorphism threshold as the infimum over all…