A ladder is a $2 \times k$ grid graph. When does a graph class $\mathcal{C}$ exclude some ladder as a minor? We show that this is the case if and only if all graphs $G$ in $\mathcal{C}$ admit a proper vertex coloring with a bounded number of colors such that for every $2$-connected subgraph $H$ of $G$, there is a color that appears exactly once in $H$. This type of vertex coloring is a relaxation of the notion of centered coloring, where for every connected subgraph $H$ of $G$, there must be a color that appears exactly once in $H$. The minimum number of colors in a centered coloring of $G$ is the treedepth of $G$, and it is known that classes of graphs with bounded treedepth are exactly those that exclude a fixed path as a subgraph, or equivalently, as a minor. In this sense, the structure of graphs excluding a fixed ladder as a minor resembles the structure of graphs without long paths. Another similarity is as follows: It is an easy observation that every connected graph with two vertex-disjoint paths of length $k$ has a path of length $k+1$. We show that every $3$-connected graph which contains as a minor a union of sufficiently many vertex-disjoint copies of a $2 \times k$ grid has a $2 \times (k+1)$ grid minor.
Our structural results have applications to poset dimension. We show that posets whose cover graphs exclude a fixed ladder as a minor have bounded dimension. This is a new step towards the goal of understanding which graphs are unavoidable as minors in cover graphs of posets with large dimension.
A graceful k-coloring of a non-empty graph $G=(V,E)$ is a proper vertex coloring $f:V(G)\rightarrow\lbrace 1,2,...,k \rbrace$, $k\geq 2$, which induces a proper edge coloring $f^{*}:E(G)\rightarrow\lbrace 1, 2, . . . , k-1 \rbrace $ defined by $f^{*}(uv) = |f(u)-f(v)|$, where $u,v\in V(G)$. The minimum $k$ for which $G$ has a graceful $k$-coloring is called graceful chromatic number, $χ_{g}(G)$. The graceful chromatic number for a few variants of ladder graphs are investigated in this article.
A $\textit{ladder}$ is a set $S \subseteq \mathbb Z^+$ such that any finite coloring of $\mathbb Z$ contains arbitrarily long monochromatic progressions with common difference in $S$. Van der Waerden's theorem famously asserts that $\mathbb Z^+$ itself is a ladder. We also discuss variants of ladders, namely $\textit{accessible}$ and $\textit{walkable}$ sets, which are sets $S$ such that any coloring of $\mathbb Z$ contains arbitrarily long (for accessible sets) or infinite (for walkable sets) monochromatic sequences with consecutive differences in $S$. We show that sets with upper density 1 a…
We introduce a new variant of graph coloring called correspondence coloring which generalizes list coloring and allows for reductions previously only possible for ordinary coloring. Using this tool, we prove that excluding cycles of lengths 4 to 8 is sufficient to guarantee 3-choosability of a planar graph, thus answering a question of Borodin.
The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of GRUNDY COLORING, the problem of determining whether a given graph has Grundy number at least $k$. We also study the variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper) and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected). We show that GRUNDY COLORING can be solved in time $O^*(2.443^n)$ and WEAK G…
Let $G$ be a graph, $r \geq t$ integers, and $N \subseteq E(G)$. An $(r,t)$-threshold-coloring of $G$ with respect to $N$ is a mapping $c: V(G) \rightarrow \{0,\ldots,r-1\}$ such that $|c(u)-c(v)| \leq t$ for every $uv \in N$ and $|c(u)-c(v)|>t$ for every $uv \in E(G) \setminus N$. A graph is total threshold colorable if there exist integers $r,t$ such that for every $N \subseteq E(G)$, $G$ admits an $(r,t)$-threshold-coloring with respect to $N$. We show that every prism is total threshold colorable, and that the Petersen graph is total threshold colorable. In contrast to this fact we show th…
Treewidth is an important graph invariant, relevant for both structural and algorithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes closed under taking induced subgraphs in which this condition is also sufficient, which we call $(tw,ω)$-bounded. Such graph classes are known to have useful algorithmic applications related to variants of the clique and $k$-coloring problems. We consider six well-known graph containment relations: the minor, topological minor, subgraph, induced minor, induced topological minor,…
A proper vertex coloring of a graph is a mapping of its vertices on a set of colors, such that two adjacent vertices are not mapped to the same color. This constraint may be interpreted in terms of the distance between to vertices and so a more general coloring concept can be defined: The strong coloring of a graph. So a k-strong coloring is a coloring where two vertices may not have the same color if their distance to each other is at most k. The 2-strong coloring of the line graph is known as the strong edge coloring. Coloring the kth power G^k of a graph G is the same as finding a k-strong …
A vertex ranking of a graph is an assignment of ranks (or colors) to the vertices of the graph, in such a way that any simple path connecting two vertices of equal rank, must contain a vertex of a higher rank. In this paper we study a relaxation of this notion, in which the requirement above should only hold for paths of some bounded length $l$ for some fixed $l$. For instance, already the case $l=2$ exhibit quite a different behavior than proper coloring. We prove upper and lower bounds on the minimum number of ranks required for several graph families, such as trees, planar graphs, graphs ex…
Half graphs and their variants, such as ladders, semi-ladders and co-matchings, are combinatorial objects that encode total orders in graphs. Works by Adler and Adler (Eur. J. Comb.; 2014) and Fabiański et al. (STACS; 2019) prove that in the powers of sparse graphs, one cannot find arbitrarily large objects of this kind. However, these proofs either are non-constructive, or provide only loose upper bounds on the orders of half graphs and semi-ladders. In this work we provide nearly tight asymptotic lower and upper bounds on the maximum order of half graphs, parameterized on the distance, in th…
For graphs $G$ and $H$, an $H$-coloring of $G$ is an edge-preserving mapping from $V(G)$ to $V(H)$. In the $H$-Coloring problem the graph $H$ is fixed and we ask whether an instance graph $G$ admits an $H$-coloring. A generalization of this problem is $H$-ColoringExt, where some vertices of $G$ are already mapped to vertices of $H$ and we ask if this partial mapping can be extended to an $H$-coloring. We study the complexity of variants of $H$-Coloring in $F$-free graphs, i.e., graphs excluding a fixed graph $F$ as an induced subgraph. For integers $a,b,c \geq 1$, by $S_{a,b,c}$ we denote the …
The Omega Institute（欧米伽研究所）；ChronoAI Pte Ltd（ChronoAI私人有限公司）；Department of Fundamental Science and Technology National Higher School of Advanced Technologies（国家高等先进技术学院基础科学与技术系）；National University of Singapore（新加坡国立大学）
We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of graphs discovered by Maffray, Trotignon and Vu\vsković. We use this method to prove the existence of elimination orderings in several classes of graphs, and to compute them in linear time. Some of the classes have already been studied, namely even-h…
We call the minimum order of any complete graph so that for any coloring of the edges by $k$ colors it is impossible to avoid a monochromatic or rainbow triangle, a Mixed Ramsey number. For any graph $H$ with edges colored from the above set of $k$ colors, if we consider the condition of excluding $H$ in the above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted $M_k(H)$. We determine this function in terms of $k$ for all colored $4$-cycles and all colored $4$-cliques. We also find bounds for $M_k(H)$ when $H$ is a monochromatic odd cycles, or a star for sufficiently large …
University of Ottawa（渥太华大学）；Jagiellonian University（雅盖隆大学）；Université libre de Bruxelles（布鲁塞尔自由大学）；École Normale Supérieure, CNRS, PSL University（巴黎高等师范学院、法国国家科学研究中心、巴黎文理研究大学）；Carleton University（卡尔顿大学）；Monash University（蒙纳士大学）
