Zdeněk Dvořák, Bernard Lidický
Let G be a planar triangle-free graph and let C be a cycle in G of length at most 8. We characterize all situations where a 3-coloring of C does not extend to a proper 3-coloring of the whole graph.
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We answer positively the question of Albertson asking whether every planar graph can be $5$-list-colored even if it contains precolored vertices, as long as they are sufficiently far apart from each other. In order to prove this claim, we also give bounds on the sizes of graphs critical with respect to 5-list coloring. In particular, if G is a planar graph, H is a connected subgraph of G and L is an assignment of lists of colors to the vertices of G such that |L(v)| >= 5 for every v in V(G)-V(H) and G is not L-colorable, then G contains a subgraph with O(|H|^2) vertices that is not L-colorable…
Given a triangle-free planar graph G and a 9-cycle C in G, we characterize situations where a 3-coloring of C does not extend to a proper 3-coloring of G. This extends previous results when C is a cycle of length at most 8.
We study 3-coloring properties of triangle-free planar graphs $G$ with two precolored 4-cycles $C_1$ and $C_2$ that are far apart. We prove that either every precoloring of $C_1\cup C_2$ extends to a 3-coloring of $G$, or $G$ contains one of two special substructures which uniquely determine which 3-colorings of $C_1\cup C_2$ extend. As a corollary, we prove that there exists a constant $D>0$ such that if $H$ is a planar triangle-free graph and $S\subseteq V(H)$ consists of vertices at pairwise distances at least $D$, then every precoloring of $S$ extends to a 3-coloring of $H$. This gives a p…
Extending edge colorings of distance-3 matchings in the Cartesian product of graphs
本文研究了在图的笛卡尔积中扩展预着色距离-3匹配的问题，证明了在满足一定条件时可以扩展为色数不超过的边着色。
Extending partial edge colorings of iterated cartesian products of cycles and paths
本文研究偶环与奇环迭代笛卡尔积的部分边着色扩展问题，在预着色边数受限或预着色边距离至少为4的条件下，证明存在与部分着色一致且色数最优的正常边着色，推广了超立方体的预着色扩展结果。
We consider the problem of extending partial edge colorings of cartesian products of graphs. More specifically, we suggest the following Evans-type conjecture: If $G$ is a graph where every precoloring of at most $k$ precolored edges can be extended to a proper $χ'(G)$-edge coloring, then every precoloring of at most $k+1$ edges of $G \square K_2$ is extendable to a proper $(χ'(G) +1)$-edge coloring of $G \square K_2$. In this paper we verify that this conjecture holds for trees, complete and complete bipartite graphs, as well as for graphs with small maximum degree. We also prove versions of …
Let G be a plane graph of girth at least five. We show that if there exists a 3-coloring phi of a cycle C of G that does not extend to a 3-coloring of G, then G has a subgraph H on O(|C|) vertices that also has no 3-coloring extending phi. This is asymptotically best possible and improves a previous bound of Thomassen. In the next paper of the series we will use this result and the attendant theory to prove a generalization to graphs on surfaces with several precolored cycles.
Extending edge-colorings of distance-2 matchings in the hypercube
本文证明了在超立方体中预着色的距离-2匹配可以扩展为合适边着色的猜想，并推广到二分图的Cartesian积结构。
In this paper, we give a polynomial time algorithm which determines if a given triangle-free graph with no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists.
In this paper, we give a polynomial time algorithm which determines if a given graph containing a triangle and no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists. In previous work, we gave a polynomial time algorithm for three-coloring triangle-free graphs with no induced seven-vertex path. Combined, our work shows that three-coloring a graph with no induced seven-vertex path can be done in polynomial time.
Dvořák, Král' and Thomas gave a description of the structure of triangle-free graphs on surfaces with respect to 3-coloring. Their description however contains two substructures (both related to graphs embedded in plane with two precolored cycles) whose coloring properties are not entirely determined. In this paper, we fill these gaps.
We give a linear-time algorithm to decide 3-colorability (and find a 3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.
We give a linear-time algorithm to decide 3-colorability of a triangle-free graph embedded in a fixed surface, and a quadratic-time algorithm to output a 3-coloring in the affirmative case. The algorithms also allow to prescribe the coloring of a bounded number of vertices.
it is shown that each triangle-free 1-planar graph with maximum degree $Δ\geq7$ can be $Δ$-colorable by Discharging Method.
A {\em total coloring} of a graph $G$ is an assignment of colors to the vertices and the edges of $G$ such that every pair of adjacent/incident elements receive distinct colors. The {\em total chromatic number} of a graph $G$, denoted by $\chiup''(G)$, is the minimum number of colors in a total coloring of $G$. The well-known Total Coloring Conjecture (TCC) says that every graph with maximum degree $Δ$ admits a total coloring with at most $Δ+ 2$ colors. A graph is {\em $1$-toroidal} if it can be drawn in torus such that every edge crosses at most one other edge. In this paper, we investigate t…
An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by Fiam{č}ik (1978) and Alon, Sudakov and Zaks (2001) states that every simple graph with maximum degree $Δ$ is acyclically edge $(Δ+ 2)$-colorable. Despite many milestones, the conjecture remains open even for planar graphs. In this paper, we confirm affirmatively the conjecture on planar graphs without intersecting triangles. We do so by first showing, by discharging methods, that every planar graph without intersecting triangles must have at le…
Let $G$ be a graph embedded on a surface $S_\varepsilon$ with Euler genus $\varepsilon > 0$, and let $P\subseteq V(G)$ be a set of vertices mutually at distance at least 4 apart. Suppose all vertices of $G$ have $H(\varepsilon)$-lists and the vertices of $P$ are precolored, where $H(\varepsilon)=\Big\lfloor\frac{7 + \sqrt{24\varepsilon + 1}}{2}\Big\rfloor$ is the Heawood number. We show that the coloring of $P$ extends to a list-coloring of $G$ and that the distance bound of 4 is best possible. Our result provides an answer to an analogous question of Albertson about extending a precoloring of…
In this paper, we prove that the class of graphs with no triangle and no induced cycle of even length at least 6 has bounded chromatic number. It is well-known that even-hole-free graphs are $χ$-bounded but we allow here the existence of $C_4$. The proof relies on the concept of Parity Changing Path, an adaptation of Trinity Changing Path which was recently introduced by Bonamy, Charbit and Thomassé to prove that graphs with no induced cycle of length divisible by three have bounded chromatic number.
Aksenov proved that in a planar graph G with at most one triangle, every precoloring of a 4-cycle can be extended to a 3-coloring of G. We give an exact characterization of planar graphs with two triangles in that some precoloring of a 4-cycle does not extend. We apply this characterization to solve the precoloring extension problem from two 4-cycles in a triangle-free planar graph in the case that the precolored 4-cycles are separated by many disjoint 4-cycles. The latter result is used in followup papers to give detailed information about the structure of 4-critical triangle-free graphs embe…
It is proved that triangle-free intersection graphs of $n$ L-shapes in the plane have chromatic number $O(\log\log n)$. This improves the previous bound of $O(\log n)$ (McGuinness, 1996) and matches the known lower bound construction (Pawlik et al., 2013).
For any fixed surface Sigma of genus g, we give an algorithm to decide whether a graph G of girth at least five embedded in Sigma is colorable from an assignment of lists of size three in time O(|V(G)|). Furthermore, we can allow a subgraph (of any size) with at most s components to be precolored, at the expense of increasing the time complexity of the algorithm to O(|V(G)|^{K(g+s)+1}) for some absolute constant K; in both cases, the multiplicative constant hidden in the O-notation depends on g and s. This also enables us to find such a coloring when it exists. The idea of the algorithm can be…
For planar graphs, we consider the problems of \emph{list edge coloring} and \emph{list total coloring}. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total coloring is the problem of coloring the edges and the vertices while ensuring that two edges that are adjacent, two vertices that are adjacent, or a vertex and an edge that are incident receive different colors. In their list extensions, instead of having the same set of colors for the whole graph, every vertex or edge is assigned some set of colors and has to b…
An acyclic edge coloring of a graph $G$ is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors in an acyclic edge coloring of $G$. It was conjectured that $\chiup'_{a}(G)\leq Δ(G) + 2$ for any simple graph $G$ with maximum degree $Δ(G)$. A graph is {\em $1$-planar} if it can be drawn on the plane such that every edge is crossed by at most one other edge. In this paper, we prove that every triangle-free $1$-planar graph $G$ has an acyclic edge coloring with $Δ(G) + 16$ color…
A celebrated result of Thomassen states that not only can every planar graph be colored properly with five colors, but no matter how arbitrary palettes of five colors are assigned to vertices, one can choose a color from the corresponding palette for each vertex so that the resulting coloring is proper. This result is referred to as 5-choosability of planar graphs. Albertson asked whether Thomassen's theorem can be extended by precoloring some vertices which are at a large enough distance apart in a graph. Here, among others, we answer the question in the case when the graph does not contain s…
A recent lower bound on the number of edges in a k-critical n-vertex graph by Kostochka and Yancey yields a half-page proof of the celebrated Grötzsch Theorem that every planar triangle-free graph is 3-colorable. In this paper we use the same bound to give short proofs of other known theorems on 3-coloring of planar graphs, among whose is the Grünbaum-Aksenov Theorem that every planar with at most three triangles is 3-colorable. We also prove the new result that every graph obtained from a triangle-free planar graph by adding a vertex of degree at most four is 3-colorable.
Let $G$ be a graph embedded in a fixed surface $Σ$ of genus $g$ and let $L=(L(v):v\in V(G))$ be a collection of lists such that either each list has size at least five, or each list has size at least four and $G$ is triangle-free, or each list has size at least three and $G$ has no cycle of length four or less. An $L$-coloring of $G$ is a mapping $ϕ$ with domain $V(G)$ such that $ϕ(v)\in L(v)$ for every $v\in V(G)$ and $ϕ(v)\neϕ(u)$ for every pair of adjacent vertices $u,v\in V(G)$. We prove * if every non-null-homotopic cycle in $G$ has length $Ω(\log g)$, then $G$ has an $L$-coloring, * if $…
Thomassen conjectured that every triangle-free planar graph on n vertices has exponentially many 3-colorings, and proved that it has at least 2^[n^(1/12)/20000] distinct 3-colorings. We show that it has at least 2^sqrt(n/362) distinct 3-colorings.
Grotzsch's theorem states that every triangle-free planar graph is 3-colorable. Several relatively simple proofs of this fact were provided by Thomassen and other authors. It is easy to convert these proofs into quadratic-time algorithms to find a 3-coloring, but it is not clear how to find such a coloring in linear time (Kowalik used a nontrivial data structure to construct an O(n log n) algorithm). We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement. As a by-prod…
We prove that every planar triangle-free graph on $n$ vertices has fractional chromatic number at most $3-\frac{1}{n+1/3}$.
A note on edge colorings distinguishing all triangles in a graph
本文研究图中区分所有三角形的边着色问题，作为区分极大星思想的扩展，估计了正常和一般着色下关于三角形的新不变量。
Thomassen conjectured that triangle-free planar graphs have exponentially many 3-colorings. Recently, he disproved his conjecture by providing examples of such graphs with $n$ vertices and at most $2^{15n/\log_2 n}$ 3-colorings. We improve his construction, giving examples of such graphs with at most $64^{n^{log_{9/2} 3}}<64^{n^{0.731}}$ 3-colorings. We conjecture this exponent is optimal.
DP-coloring, also known as correspondence coloring, is introduced by Dvo{ř}{á}k and Postle. It is a generalization of list coloring. In this paper, we show that every connected toroidal graph without triangles adjacent to $5$-cycles has minimum degree at most three unless it is a 2-connected $4$-regular graph with Euler characteristic $ε(G) = 0$. Consequently, every toroidal graph without triangles adjacent to $5$-cycles is DP-$4$-colorable. In the final, we show that every planar graph without two certain subgraphs is DP-$4$-colorable. As immediate consequences, (i) every planar graph without…
We show that if a coloring of the plane has the properties that any two points at distance one are colored differently and the plane is partitioned into uniformly colored triangles under certain conditions, then it requires at least seven colors. This is also true for a coloring using uniformly colored polygons if it has a point bordering at least four polygons.
Better 3-coloring algorithms: excluding a triangle and a seven vertex path
本文提出一种针对$\{P_7,C_3\}$-free图的列表3-着色算法，复杂度优于现有方法；并证明当$t \leq 7$时该类问题的极小障碍有限，从而推出多项式时间证书算法。
We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles are at distance at least d, then G is 3-colorable. In fact, we prove a more general theorem. Let G be a planar graph, and let H be a set of connected subgraphs of G, each of bounded size, such that every two distinct members of H are at least a specified distance apart and all triangles of G are contained in \bigcup{H}. We give a sufficient condition for the existence of a 3-coloring phi of G such that for every B\in H, the restriction of phi t…
Recently, it was proved that triangle-free intersection graphs of $n$ line segments in the plane can have chromatic number as large as $Θ(\log\log n)$. Essentially the same construction produces $Θ(\log\log n)$-chromatic triangle-free intersection graphs of a variety of other geometric shapes---those belonging to any class of compact arc-connected sets in $\mathbb{R}^2$ closed under horizontal scaling, vertical scaling, and translation, except for axis-parallel rectangles. We show that this construction is asymptotically optimal for intersection graphs of boundaries of axis-parallel rectangles…
In the problem of 2-coloring without monochromatic triangles (or triangle-tree 2-coloring), vertices of the simple, connected, undirected graph are colored with either 'black' or 'white' such that there are no 3 mutually adjacent vertices of the same color. In this paper we are positively answering the question posed in our previous work, namely, if there exists an algorithm solving 2-coloring without monochromatic triangles on planar graphs with linear-time complexity.
A note on transformations of edge colorings of chordless graphs and triangle-free graphs
本文指出通过对Bonamy等人（2023）关于无三角形图边着色Kempe变换结论的证明进行微小修改，可将该结论推广到包含无三角形图和无弦图的更大类简单图中。
We give a full, correct proof of the following result, earlier claimed by Erdős and Komjáth. If the Continuum Hypothesis holds then there is a coloring of the plane with countably many colors, with no monocolored right triangle.
For positive integers $n$ and $r$, we consider $n$-vertex graphs with the maximum number of $r$-edge-colorings with no copy of a triangle where exactly two colors appear. We prove that, if $2 \leq r \leq 26$ and $n$ is sufficiently large, the maximum is attained by the bipartite Turán graph $T_2(n)$ on $n$ vertices. This is best possible, as $T_2(n)$ is not extremal for $r \geq 27$ colors and $n \geq 3$.
Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of non-empty sets. By an $L$-coloring of $G$ we mean a (proper) coloring $ϕ$ of $G$ such that $ϕ(v)\in L(v)$ for every vertex $v$ of $G$. Thomassen proved that if $v_1,v_2\in V(C)$ are adjacent, $L(v_1)\ne L(v_2)$, $|L(v)|\ge3$ for every $v\in V(C)-\{v_1,v_2\}$ and $|L(v)|\ge5$ for every $v\in V(G)-V(C)$, then $G$ has an $L$-coloring. What happens when $v_1$ and $v_2$ are not adjacent? Then an $L$-coloring need not exist, but in the first paper of this series we have shown that it exists if $|L(v_1)|,|L(v_2)|\…
In the first partial result toward Steinberg's now-disproved three coloring conjecture, Abbott and Zhou used a counting argument to show that every planar graph without cycles of lengths 4 through 11 is 3-colorable. Implicit in their proof is a fact about plane graphs: in any plane graph of minimum degree 3, if no two triangles share an edge, then triangles make up strictly less than 2/3 of the faces. We show how this result, combined with Kostochka and Yancey's resolution of Ore's conjecture for k = 4, implies that every planar graph without cycles of lengths 4 through 8 is 3-colorable.
Thomassen conjectured that triangle-free planar graphs have an exponential number of $3$-colorings. We show this conjecture to be equivalent to the following statement: there exists a positive real $α$ such that whenever $G$ is a planar graph and $A$ is a subset of its edges whose deletion makes $G$ triangle-free, there exists a subset $A'$ of $A$ of size at least $α|A|$ such that $G-(A\setminus A')$ is $3$-colorable. This equivalence allows us to study restricted situations, where we can prove the statement to be true.
Variations on Ramsey numbers and minimum numbers of monochromatic triangles in line $2$-colorings of configurations
本文探索了拉姆齐数与单色化配置中单色三角形的最小数变体，通过定义线2-着色配置并计算具体对称示例，扩展并完善了相关图论理论。
Coloring equilateral triangles
本研究在集合论框架下证明，相对于不可达基数，ZF+DC成立时欧氏空间等边三角形超图有可数色数，但平面等腰三角形超图不具备该性质，推进了几何超图着色与集合论相容性研究。
By a theorem of Johansson, every triangle-free graph $G$ of maximum degree $Δ$ has chromatic number at most $(C+o(1))Δ/\log Δ$ for some universal constant $C > 0$. Using the entropy compression method, Molloy proved that one can in fact take $C = 1$. Here we show that for every $q \geq (1 + o(1))Δ/\log Δ$, the number $c(G,q)$ of proper $q$-colorings of $G$ satisfies $c(G, q) \,\geq\, \left(1 - \frac{1}{q}\right)^m ((1-o(1))q)^n$, where $n = |V(G)|$ and $m = |E(G)|$. Except for the $o(1)$ term, this lower bound is best possible as witnessed by random $Δ$-regular graphs. When $q = (1 + o(1)) Δ/\…
We show that the size of a 4-critical graph of girth at least five is bounded by a linear function of its genus. This strengthens the previous bound on the size of such graphs given by Thomassen. It also serves as the basic case for the description of the structure of 4-critical triangle-free graphs embedded in a fixed surface, presented in a future paper of this series.
Thomassen proved that any plane graph of girth 5 is list-colorable from any list assignment such that all vertices have lists of size two or three and the vertices with list of size two are all incident with the outer face and form an independent set. We present a strengthening of this result, relaxing the constraint on the vertices with list of size two. This result is used to bound the size of the 3-list-coloring critical plane graphs with one precolored face.
We give a randomized algorithm that properly colors the vertices of a triangle-free graph G on n vertices using O(Δ(G)/ log Δ(G)) colors, where Δ(G) is the maximum degree of G. The algorithm takes O(n\Delta2(G)logΔ(G)) time and succeeds with high probability, provided Δ(G) is greater than log^{1+ε}n for a positive constant ε. The number of colors is best possible up to a constant factor for triangle-free graphs. As a result this gives an algorithmic proof for a sharp upper bound of the chromatic number of a triangle-free graph, the existence of which was previously established by Kim and Johan…
DP-coloring (also known as correspondence coloring) of a simple graph is a generalization of list coloring. It is known that planar graphs without 4-cycles adjacent to triangles are 4-choosable, and planar graphs without 4-cycles are DP-4-colorable. In this paper, we show that planar graphs without 4-cycles adjacent to triangles are DP-4-colorable, which is an extension of the two results above.
Coloring triangles in graphs
本文研究图中三角形着色的定量性质，定义诱导 Ramsey 数 $R_{\ ext{ind}}^Δ(F)$，并针对若干特定图类研究其大小，以改进已有的塔型上界。
Let $H$ be a graph with $Δ(H) \leq 2$, and let $G$ be obtained from $H$ by gluing in vertex-disjoint copies of $K_4$. We prove that if $H$ contains at most one odd cycle of length exceeding $3$, or if $H$ contains at most $3$ triangles, then $χ(G) \leq 4$. This proves the Strong Coloring Conjecture for such graphs $H$. For graphs $H$ with $Δ=2$ that are not covered by our theorem, we prove an approximation result towards the conjecture.
An {\em acyclic edge coloring} of a graph $G$ is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The {\em acyclic chromatic index} $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors needed in an acyclic edge coloring of $G$. Fiamčík (1978) conjectured that $\chiup_{a}'(G) \leq Δ(G) + 2$, where $Δ(G)$ is the maximum degree of $G$. This conjecture is well known as Acyclic Edge Coloring Conjecture (AECC). A graph $G$ with maximum degree at most $κ$ is {\em $κ$-deletion-minimal} if $\ch…
A Gallai-coloring (Gallai-$k$-coloring) is an edge-coloring (with colors from $\{1, 2, \ldots, k\}$) of a complete graph without rainbow triangles. Given a graph $H$ and a positive integer $k$, the $k$-colored Gallai-Ramsey number $GR_k(H)$ is the minimum integer $n$ such that every Gallai-$k$-coloring of the complete graph $K_n$ contains a monochromatic copy of $H$. In this paper, we consider two extremal problems related to Gallai-$k$-colorings. First, we determine upper and lower bounds for the maximum number of edges that are not contained in any rainbow triangle or monochromatic triangle …
Precoloring 3-extension on outerplanar graphs
本文研究了外平面图的预着色扩展问题，证明了在特定条件下，预着色可以扩展为图的3着色。
A triangle in a hypergraph is a collection of distinct vertices u,v,w and distinct edges e,f,g with u,v \in e, v,w \in f, w,u \in g, and \{u,v,w\} \cap e \cap f \cap g=\emptyset. The i-degree of a vertex in a hypergraph is the number of edges of size i containing it. We prove that every triangle-free hypergraph of rank three (edges have size two or three) with maximum 3-degree Δ_3 and maximum 2-degree Δ_2 has list chromatic number at most c max{Δ_2/ log{Δ_2}}, (Δ_3 / log{Δ_3})^(1/2)} for some absolute positive constant c. This generalizes a result of Johansson and a result of Frieze and the se…
Let G be a 4-critical graph with t triangles, embedded in a surface of genus g. Let c be the number of 4-cycles in G that do not bound a 2-cell face. We prove that the sum of lengths of (>=5)-faces of G is at most linear in g+t+c-1.
We study a certain relaxation of the classic vertex coloring problem, namely, a coloring of vertices of undirected, simple graphs, such that there are no monochromatic triangles. We give the first classification of the problem in terms of classic and parametrized algorithms. Several computational complexity results are also presented, which improve on the previous results found in the literature. We propose the new structural parameter for undirected, simple graphs -- the triangle-free chromatic number $χ_3$. We bound $χ_3$ by other known structural parameters. We also present two classes of g…
Fox--Grinshpun--Pach showed that every $3$-coloring of the complete graph on $n$ vertices without a rainbow triangle contains a clique of size $Ω\left(n^{1/3}\log^2 n\right)$ which uses at most two colors, and this bound is tight up to the constant factor. We show that if instead of looking for large cliques one only tries to find subgraphs of large chromatic number, one can do much better. We show that every such coloring contains a $2$-colored subgraph with chromatic number at least $n^{2/3}$, and this is best possible. We further show that for fixed positive integers $s,r$ with $s\leq r$, e…
We show that triangle-free penny graphs have degeneracy at most two, list coloring number (choosability) at most three, diameter $D=Ω(\sqrt n)$, and at most $\min\bigl(2n-Ω(\sqrt n),2n-D-2\bigr)$ edges.
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