相关论文: The choice number versus the chromatic number for …
Erd\"os conjectured that if $G$ is a triangle free graph of chromatic number at least $k\geq 3$, then it contains an odd cycle of length at least $k^{2-o(1)}$ \cite{sudakovverstraete, verstraete}. Nothing better than a linear bound…
The b-chromatic number of a graph $G$, denoted by $\phi(G)$, is the largest integer $k$ that $G$ admits a proper $k$-coloring such that each color class has a vertex that is adjacent to at least one vertex in each of the other color…
In 1967, Erd\H{o}s asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erd\H{o}s and Hajnal together with Shearer's classical upper bound for the off-diagonal Ramsey number $R(3,…
The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same…
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 adaptable choosability of a multigraph $G$, denoted $\mathrm{ch}_a(G)$, is the smallest integer $k$ such that any edge labelling, $\tau$, of $G$ and any assignment of lists of size $k$ to the vertices of $G$ permits a list colouring,…
The $b$-chromatic number of a graph $G$, denoted by $b(G)$, is the largest positive integer $k$ such that there exists a proper coloring for G with $k$ colors in which every color class contains at least one vertex adjacent to some vertex…
The local chromatic number of a graph was introduced by Erd\H{o}s et al. [4]. In [17] a connection to topological properties of (a box complex of) the graph was established and in [18] it was shown that if a graph is strongly topologically…
An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that for any vertex $v$, there are at least $\min\{r, deg_G(v) \}$ distinct colors in $N_G(v)$. The $r$-dynamic chromatic number $\chi_r^d(G)$ of a graph $G$ is the…
A number of famous results about triangle-free graphs determine the maximum chromatic number of graphs of order n with minimum degree greater than n/3. In this note these results are extended to Kr-free graphs.
We determine the list chromatic number of the square of a graph $\chil(G^2)$ in terms of its maximum degree $\Delta$ when its maximum average degree, denoted $\mad(G)$, is sufficiently small. For $\Delta\ge 6$, if…
Chromatic choosability is a notion of fundamental importance in list coloring. A graph $G$ is chromatic-choosable when its chromatic number, $\chi(G)$, is equal to its list chromatic number $\chi_{\ell}(G)$. In 1990, Kostochka and Sidorenko…
A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of non-negative integers (with minimum assignment $0$) to its vertices such that the adjacent vertices must receive integers at least two apart and vertices at distance…
In this paper we find chromatic numbers of distance graphs $G(n,3,2)$ for infinitely many n. Also we improve upper bound for $\chi(G(n,r,s))$ in large part of cases.
Oriented chromatic number of an oriented graph $G$ is the minimum order of an oriented graph $H$ such that $G$ admits a homomorphism to $H$. The oriented chromatic number of an unoriented graph $G$ is the maximal chromatic number over all…
We give an exact characterization of 3-colorability of triangle-free graphs drawn in the torus, in the form of 186 "templates" (graphs with certain faces filled by arbitrary quadrangulations) such that a graph from this class is not…
It is known (Bollob\'{a}s (1978); Kostochka and Mazurova (1977)) that there exist graphs of maximum degree $\Delta$ and of arbitrarily large girth whose chromatic number is at least $c \Delta / \log \Delta$. We show an analogous result for…
Let $f_{o}(G)$ be the maximum order of an odd induced subgraph of $G$. In 1992, Scott proposed a conjecture that $f_{o}(G)\geq \frac {n} {2\chi(G)}$ for a graph $G$ of order $n$ without isolated vertices, where $\chi(G)$ is the chromatic…
Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented…
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…