Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges is in each color class. If there exists an integer $k$ such that, for $n$ sufficiently large, every 2-coloring of $K_n$ with more than $k$ edges in each color contains a balanced copy of $G$, then we say that $G$ is balanceable. The smallest integer $k$ such that this holds is called the balancing number of $G$.
In this paper, we define a more general variant of the balancing number, the list balancing number, by considering 2-list edge colorings of $K_n$, where every edge $e$ has an associated list $L(e)$ which is a nonempty subset of the color set $\{r,b\}$. In this case, edges $e$ with $L(e) = \{r,b\}$ act as jokers in the sense that their color can be chosen $r$ or $b$ as needed. In contrast to the balancing number, every graph has a list balancing number. Moreover, if the balancing number exists, then it coincides with the list balancing number.
We give the exact value of the list balancing number for all cycles except for $4k$-cycles for which we give tight bounds. In addition, we give general bounds for the list balancing number of non-balanceable graphs based on the extremal number of its subgraphs, and study the list balancing number of $K_5$, which turns out to be surprisingly large.
A {\em strong edge coloring} of a graph is a proper edge coloring in which every color class is an induced matching. The {\em strong chromatic index} of a graph is the minimum number of colors needed to obtain a strong edge coloring. In an analogous way, we can define the list version of strong edge coloring and list version of strong chromatic index. In this paper, we prove that if $G$ is a graph with maximum degree at most four and maximum average degree less than $3$, then the list strong chromatic index is at most $3Δ+ 1$, where $Δ$ is the maximum degree of $G$. In addition, we prove that …
For positive integers $a$ and $b$, a graph $G$ is $(a:b)$-choosable if, for each assignment of lists of $a$ colors to the vertices of $G,$ each vertex can be colored with a set of $b$ colors from its list so that adjacent vertices are colored with disjoint sets. We show that for positive integers $a$ and $b$, every bipartite planar graph is $(a:b)$-choosable iff $\frac{a}{b} \ge 3$. For general planar graphs, we show that if $\frac{a}{b} < 4\frac{2}{5}$, then there exists a planar graph that is not $(a:b)$-choosable, thus improving on a result of X. Zhu, which had $4\frac{2}{9}$. Lastly, we sh…
Lists of equivalence classes of words under rotation or rotation plus reversal (i.e., necklaces and bracelets) have many uses, and efficient algorithms for generating these lists exist. In combinatorial group theory elements of a group are typically written as words in the generators and their inverses, and necklaces and bracelets correspond to conjugacy classes and relators respectively. We present algorithms to generate lists of freely and cyclically reduced necklaces and bracelets in free groups. Experimental evidence suggests that these algorithms are CAT -- that is, they run in constant a…
The clustered chromatic number of a graph class is the minimum integer $t$ such that for some $C$ the vertices of every graph in the class can be colored in $t$ colors so that every monochromatic component has size at most $C$. We show that the clustered chromatic number of the class of graphs embeddable on a given surface is four, proving the conjecture of Esperet and Ochem. Additionally, we study the list version of the concept and characterize the minor-closed classes of graphs of bounded treewidth with given clustered list chromatic number. We further strengthen the above results to solve …
A graph is (7, 2)-edge-choosable if, for every assignment of lists of size 7 to the edges, it is possible to choose two colors for each edge from its list so that no color is chosen for two incident edges. We show that every 3-edge-colorable graph is (7, 2)-edge-choosable and also that many non-3-edge-colorable 3-regular graphs are (7, 2)-edge-choosable.
In 1977, Wegner conjectured that the chromatic number of the square of every planar graph $G$ with maximum degree $Δ\ge8$ is at most $\bigl\lfloor\frac32Δ\bigr\rfloor+1$. We show that it is at most $\frac32 Δ(1+o(1))$ (where the $o(1)$ is as $Δ\to+\infty$), and indeed that this is true for the list chromatic number and for more general classes of graphs.
We present two algorithms to list certain classes of monomino-domino coverings which conform to the \emph{tatami} restriction; no four tiles meet. Our methods exploit structural features of tatami coverings in order to create the lists in $O(1)$ time per covering. This is faster than known methods for generating certain classes of matchings in bipartite graphs. We discuss tatami coverings of $n\times n$ grids with $n$ monominoes and $v$ vertical dominoes, as well as tatami coverings of a two-way infinitely-wide strip of constant height, subject to the constraint that they have a finite number …
Bernshteyn and Lee defined a new notion, weak degeneracy, which is slightly weaker than the ordinary degeneracy. It is proved that strictly $f$-degenerate transversal is a common generalization of list coloring, $L$-forested-coloring and DP-coloring. In this paper, we consider three classes of graphs, including planar graphs without any configuration in Fig. 2, toroidal graphs without any configuration in Fig. 5, and planar graphs without intersecting $5$-cycles. We give structural results for each class of graphs, and prove each structure is reducible for weakly $3$-degenerate and the existen…
DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvořák and Postle (2017). In this paper, we prove that every planar graph $G$ without $4$-cycles adjacent to $k$-cycles is DP-$4$-colorable for $k=5$ and $6$. As a consequence, we obtain two new classes of $4$-choosable planar graphs. We use identification of verticec in the proof, and actually prove stronger statements that every pre-coloring of some short cycles can be extended to the whole graph.
In this paper, we study the group and list group colorings of total graphs and we give two group versions of the total and list total colorings conjectures. We establish the group version of the total coloring conjecture for the following classes of graphs: graphs with small maximum degree, two-degenerate graphs, planner graphs with maximum degree at least 11, planner graphs without certain small cycles, outerplanar and near-outerplanar graphs. In addition, the group version of the list total coloring conjecture is established for forests, outerplanar graphs and graphs with maximum degree at m…
Signed list edge coloring in graphs of bounded treewidth
Yan’an University（延安大学）；School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）；School of Mathematics and Statistics & Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), Fujian Normal University（福建师范大学数学与统计学院与分析数学及应用重点实验室（教育部））；School of Marine Science and Technology, Northwestern Polytechnical University（西北工业大学海洋科学与技术学院）
It was noted already in the 90s that many classic graph classes, such as interval, chordal, and bipartite graphs, can be characterized by the existence of an ordering of the vertices avoiding some ordered subgraphs, called patterns. Very recently, all the classes corresponding to patterns on three vertices (including the ones mentioned above) have been listed, and proved to be efficiently recognizable. In contrast, very little is known about patterns on four vertices. One of the few graph classes characterized by a pattern on four vertices is the class of intersection graphs of rectangles that…
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 $(γ,μ)$-coloring, where the color lists have sequential values with known lower and upper bounds. This work discusses the choosability property, that consists in determining the least number $k$ for which it has a proper list coloring no matter how one assigns a list of $k$ colors to each vertex. This is a $Π_2^P$-complete problem, however, we show that $k$-$(γ,μ)$-choosability is an …
The complexity of graph isomorphism (GraphIso) is a famous unresolved problem in theoretical computer science. For graphs $G$ and $H$, it asks whether they are the same up to a relabeling of vertices. In 1981, Lubiw proved that list restricted graph isomorphism (ListIso) is NP-complete: for each $u \in V(G)$, we are given a list ${\mathfrak L}(u) \subseteq V(H)$ of possible images of $u$. After 35 years, we revive the study of this problem and consider which results for GraphIso translate to ListIso. We prove the following: 1) When GraphIso is GI-complete for a class of graphs, it translates i…
In this note we present House of Graphs (http://hog.grinvin.org) which is a new database of graphs. The key principle is to have a searchable database and offer -- next to complete lists of some graph classes -- also a list of special graphs that already turned out to be interesting and relevant in the study of graph theoretic problems or as counterexamples to conjectures. This list can be extended by users of the database.
In the list coloring problem for two matroids, we are given matroids $M_1=(S,{\cal I}_1)$ and $M_2=(S,{\cal I}_2)$ on the same ground set $S$, and the goal is to determine the smallest number $k$ such that given arbitrary lists $L_s$ of $k$ colors for $s\in S$, it is possible to choose a color from each list so that every monochromatic set is independent in both $M_1$ and $M_2$. When both $M_1$ and $M_2$ are partition matroids, Galvin's list coloring theorem for bipartite graphs gives the answer. One of the main open questions is to decide if there exists a constant $c$ such that if the colori…
A graph is $\ell$-choosable if, for any choice of lists of $\ell$ colors for each vertex, there is a list coloring, which is a coloring where each vertex receives a color from its list. We study complexity issues of choosability of graphs when the number $k$ of colors is limited. We get results which differ surprisingly from the usual case where $k$ is implicit and which extend known results for the usual case. We also exhibit some classes of graphs (defined by structural properties of their blocks) which are choosable.
For a graph $F$, a graph $G$ is \emph{$F$-free} if it does not contain an induced subgraph isomorphic to $F$. For two graphs $G$ and $H$, an \emph{$H$-coloring} of $G$ is a mapping $f:V(G)\rightarrow V(H)$ such that for every edge $uv\in E(G)$ it holds that $f(u)f(v)\in E(H)$. We are interested in the complexity of the problem $H$-{\sc Coloring}, which asks for the existence of an $H$-coloring of an input graph $G$. In particular, we consider $H$-{\sc Coloring} of $F$-free graphs, where $F$ is a fixed graph and $H$ is an odd cycle of length at least 5. This problem is closely related to the we…
