相关论文: On-Line Choice Number of Complete Multipartite Gra…
A graph $G$ is called \emph{chromatic-choosable} if its choice number is equal to its chromatic number, namely $Ch(G)=\chi(G)$. Ohba has conjectured that every graph $G$ satisfying $|V(G)|\leq 2\chi(G)+1$ is chromatic-choosable. Since each…
The on-line choice number of a graph is a variation of the choice number defined through a two person game. It is at least as large as the choice number for all graphs and is strictly larger for some graphs. In particular, there are graphs…
We calculate the asymptotic value of the choice number of complete multi-partite graphs.
In this paper, the on-line list colouring of binomial random graphs G(n,p) is studied. We show that the on-line choice number of G(n,p) is asymptotically almost surely asymptotic to the chromatic number of G(n,p), provided that the average…
Proportional choosability is a list analogue of equitable coloring that was introduced in 2019. The smallest $k$ for which a graph $G$ is proportionally $k$-choosable is the proportional choice number of $G$, and it is denoted…
We study the paintability, an on-line version of choosability, of complete multipartite graphs. We do this by considering an equivalent chip game introduced by Duraj, Gutowski, and Kozik. We consider complete multipartite graphs with $ n $…
The problem of determining if the on-line chromatic number of a graph is less than or equal to k, given a pre-coloring, is shown to be PSPACE-complete.
The Hall number is a graph parameter closely related to the choice number. Here it is shown that the Hall numbers of the complete multipartite graphs $K(m,2,\ldots,2)$, $m\ge 2$, are equal to their choice numbers.
In this paper we investigate the game chromatic number for complete multipartite graphs. We devise several strategies for Alice, and one strategy for Bob, and we prove their optimality in all complete multipartite graphs with no singletons.…
A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not necessarily $(c:d)$-choosable. The simplest case of another problem, stated by the same…
The main goal of this paper is to formalize and explore a connection between chromatic properties of graphs with geometric representations and competitive analysis of on-line algorithms, which became apparent after the recent construction…
We show that there is a constant $k$ such that when $r \geq 2$ and $m \geq r^k$, the complete $r$-partite graph $K_{m*r}$ has a non-colorable list assignment $L$ such that $|L(v)| \geq \frac{7}{750}r\ln m$ for all $v$ and such that $|L(u)…
A graph is k-total colourable if there is an assignment of k different colours to the vertices and edges of the graph such that no two adjacent nor incident elements receive the same colour. The total chromatic number of some direct product…
A $k$-uniform hypergraph (or $k$-graph) $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that each edge in $E$ contains precisely one vertex from each $V_i$. We show that $k$-partite $k$-graphs of…
In the online graph coloring problem, vertices from a graph G, known in advance, arrive in an online fashion and an algorithm must immediately assign a color to each incoming vertex v so that the revealed graph is properly colored. The…
A graph $G=(V,E)$ is called $(k,k')$-choosable if for any total list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a mapping…
Let $\mathrm{ch}(G)$ denote the choice number of a graph $G$, and let $K_{s*k}$ be the complete $k$-partite graph with $s$ vertices in each part. Erd\H{o}s, Rubin, and Taylor showed that $\mathrm{ch}( K_{2*k})=k$, and suggested the problem…
The list coloring problem is a variation of the classical vertex coloring problem, extensively studied in recent years, where each vertex has a restricted list of allowed colors, and having some variations as the $(\gamma,\mu)$-coloring,…
We introduce a variant of the well-studied sum choice number of graphs, which we call the interactive sum choice number. In this variant, we request colours to be added to the vertices' colour-lists one at a time, and so we are able to make…
The $k$th power $G^k$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^k$ if the distance between $u$ and $v$ in $G$ is at most $k$. Let $\chi(H)$ and $\chi_l(H)$ be the chromatic number…