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.
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.
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.
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.
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 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.
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.
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).
Let G be a plane graph with exactly one triangle T and all other cycles of length at least 5, and let C be a facial cycle of G of length at most six. We prove that a 3-coloring of C does not extend to a 3-coloring of G if and only if C has length exactly six and there is a color x such that either G has an edge joining two vertices of C colored x, or T is disjoint from C and every vertex of T is adjacent to a vertex of C colored x. This is a lemma to be used in a future paper of this series.
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…
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 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.
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…
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…
An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by $a'(G)$. It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that $a'(G)\le Δ+2$, where $Δ=Δ(G)$ denotes the maximum degree of the graph. If every induced subgraph $H$ of $G$ satisfies the condition $\vert E(H) \vert \le 2\vert V(H) \vert -1$, we say that the graph $G$ satisfies $Property\ A$. In this paper, we prove that…
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/\…
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…
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.
A graph drawn in a surface is a near-quadrangulation if the sum of the lengths of the faces different from 4-faces is bounded by a fixed constant. We leverage duality between colorings and flows to design an efficient algorithm for 3-precoloring-extension in near-quadrangulations of orientable surfaces. Furthermore, we use this duality to strengthen previously known sufficient conditions for 3-colorability of triangle-free graphs drawn in orientable surfaces.
Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G is triangle-free and all lists have size at least four, then there exists an L-coloring respecting at least a constant fraction of the preferences.
A proper edge coloring of a graph $G$ is called acyclic if there is no bichromatic cycle in $G$. The acyclic chromatic index of $G$, denoted by $χ'_a(G)$, is the least number of colors $k$ such that $G$ has an acyclic edge $k$-coloring. The maximum average degree of a graph $G$, denoted by $\mad(G)$, is the maximum of the average degree of all subgraphs of $G$. In this paper, it is proved that if $\mad(G)<4$, then $χ'_a(G)\leq{Δ(G)+2}$; if $\mad(G)<3$, then $χ'_a(G)\leq{Δ(G)+1}$. This implies that every triangle-free planar graph $G$ is acyclically edge $(Δ(G)+2)$-colorable.
In 1964 Vizing proved that starting from any k-edge-coloring of a graph G one can reach, using only Kempe swaps, a ($Δ$ + 1)-edge-coloring of G where $Δ$ is the maximum degree of G. One year later he conjectured that one can also reach a $Δ$-edge-coloring of G if there exists one. Bonamy et. al proved that the conjecture is true for the case of triangle-free graphs. In this paper we prove the conjecture for all graphs.
We study the \emph{maximum differential coloring problem}, where the vertices of an $n$-vertex graph must be labeled with distinct numbers ranging from $1$ to $n$, so that the minimum absolute difference between two labels of any two adjacent vertices is maximized. As the problem is \NPH for general graphs~\cite{leung1984}, we consider planar graphs and subclasses thereof. We initially prove that the maximum differential coloring problem remains \NPH, even for planar graphs. Then, we present tight bounds for regular caterpillars and spider graphs. Using these new bounds, we prove that the Mill…
Felsner, Hurtado, Noy and Streinu (2000) conjectured that arrangement graphs of simple great-circle arrangements have chromatic number at most $3$. Motivated by this conjecture, we study the colorability of arrangement graphs for different classes of arrangements of (pseudo-)circles. In this paper the conjecture is verified for $\triangle$-saturated pseudocircle arrangements, i.e., for arrangements where one color class of the 2-coloring of faces consists of triangles only, as well as for further classes of (pseudo-)circle arrangements. These results are complemented by a construction which ma…
We consider the class A of graphs that contain no odd hole, no antihole, and no ``prism'' (a graph consisting of two disjoint triangles with three disjoint paths between them). We show that the coloring algorithm found by the second and fourth author can be implemented in time O(n^2m) for any graph in A with n vertices and m edges, thereby improving on the complexity proposed in the original paper.
The computational complexity of the Vertex Coloring problem is known for all hereditary classes of graphs defined by forbidding two connected five-vertex induced subgraphs, except for seven cases. We prove the polynomial-time solvability of four of these problems: for ($P_5$, dart)-free graphs, ($P_5$, banner)-free graphs, ($P_5$, bull)-free graphs, and (fork, bull)-free graphs.
It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without an induced path on $t$ vertices, computes a coloring with $\max\{5,2\lceil{\frac{t-1}{2}}\rceil-2\}$ many colors. If the input graph is triangle-free, we only need $\max\{4,\lceil{\frac{t-1}{2}}\rceil+1\}$ many colors. The running time of our algorithm is $O((3^{t-2}+t^2)m+n)$ if the input graph has $n$ vertices and $m$ edges.
A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an \emph{interval cyclic $t$-coloring} if all colors are used, and the edges incident to each vertex $v\in V(G)$ are colored by $d_{G}(v)$ consecutive colors modulo $t$, where $d_{G}(v)$ is the degree of a vertex $v$ in $G$. A graph $G$ is \emph{interval cyclically colorable} if it has an interval cyclic $t$-coloring for some positive integer $t$. The set of all interval cyclically colorable graphs is denoted by $\mathfrak{N}_{c}$. For a graph $G\in \mathfrak{N}_{c}$, the least and the greatest values of $t$ for which it …
We study the exact square chromatic number of subcubic planar graphs. An exact square coloring of a graph G is a vertex-coloring in which any two vertices at distance exactly 2 receive distinct colors. The smallest number of colors used in such a coloring of G is its exact square chromatic number, denoted $χ^{\sharp 2}(G)$. This notion is related to other types of distance-based colorings, as well as to injective coloring. Indeed, for triangle-free graphs, exact square coloring and injective coloring coincide. We prove tight bounds on special subclasses of planar graphs: subcubic bipartite pla…
Several classical constructions illustrate the fact that the chromatic number of a graph can be arbitrarily large compared to its clique number. However, until very recently, no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ in $\mathbb{R}^2$ that is not an axis-aligned rectangle and for any positive integer $k$ produces a family $\mathcal{F}$ of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$, such that no three sets in $\mathcal{F}$…
Erdös 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 (\cite{gyarfas}, Problem 5.1.55 in \cite{West}) was so far known. We make progress on this conjecture by showing that $G$ contains an odd cycle of length at least $O(k\log\log k)$. Erdös' conjecture is known to hold for graphs with girth at least 5. We show that if a girth 4 graph is $C_5$ free, then Erdös' conjecture holds. When the number of vertices is not too larg…
A recent palette sparsification theorem of Assadi, Chen, and Khanna [SODA'19] states that in every $n$-vertex graph $G$ with maximum degree $Δ$, sampling $O(\log{n})$ colors per each vertex independently from $Δ+1$ colors almost certainly allows for proper coloring of $G$ from the sampled colors. Besides being a combinatorial statement of its own independent interest, this theorem was shown to have various applications to design of algorithms for $(Δ+1)$ coloring in different models of computation on massive graphs such as streaming or sublinear-time algorithms. In this paper, we further study…
An injective $k$-edge-coloring of a graph $G$ is an assignment of colors, i.e. integers in $\{1, \ldots , k\}$, to the edges of $G$ such that any two edges each incident with one distinct endpoint of a third edge, receive distinct colors. The problem of determining whether such a $k$-coloring exists is called k-INJECTIVE EDGE-COLORING. We show that 3-INJECTIVE EDGE-COLORING is NP-complete, even for triangle-free cubic graphs, planar subcubic graphs of arbitrarily large girth, and planar bipartite subcubic graphs of girth~6. 4-INJECTIVE EDGE-COLORING remains NP-complete for cubic graphs. For an…
In spite of the extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of the 3-coloring problem on graphs with small diameter, i.e. with diameter 2 or 3, has been a longstanding and challenging open question. For graphs with diameter 2 we provide the first subexponential algorithm with complexity $2^{O(\sqrt{n\log n})}$, which is asymptotically the same as the currently best known time complexity for the graph isomorphism (GI) problem. Moreover, we prove that the graph isomorphism problem on 3-colorable graphs with diameter 2 is GI-complete…
Coloring problems in graphs have been used to model a wide range of real applications. In particular, the List Coloring Problem generalizes the well-known Graph Coloring Problem for which many exact algorithms have been developed. In this work, we present a Branch-and-Price algorithm for the weighted version of the List Coloring Problem, based on the one developed by Mehrotra and Trick (1996) for the Graph Coloring Problem. This version considers non-negative weights associated to each color and it is required to assign a color to each vertex from predetermined lists in such a way the sum of w…
For an integer $r>0$, a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to vertices with at least $min\{r, d(v)\}$ different colors. The smallest integer $k$ for which a graph $G$ has a conditional $(k,r)$-coloring is called the $r$th order conditional chromatic number, denoted by $χ_r(G)$. It is easy to see that the conditional coloring is a generalization of the traditional vertex coloring for which $r=1$. In this paper, we consider the complexity of the conditional colorings of graphs.…
In 1994, Thomassen proved that every planar graph is 5-list-colorable. In 1995, Thomassen proved that every planar graph of girth at least five is 3-list-colorable. His proofs naturally lead to quadratic-time algorithms to find such colorings. Here, we provide the first such linear-time algorithms to find such colorings. For a fixed surface S, Thomassen showed in 1997 that there exists a linear-time algorithm to decide if a graph embedded in S is 5-colorable and similarly in 2003 if a graph of girth at least five embedded in S is 3-colorable. Using the theory of hyperbolic families, the author…
The spectral independence approach of Anari et al. (2020) utilized recent results on high-dimensional expanders of Alev and Lau (2020) and established rapid mixing of the Glauber dynamics for the hard-core model defined on weighted independent sets. We develop the spectral independence approach for colorings, and obtain new algorithmic results for the corresponding counting/sampling problems. Let $α^*\approx 1.763$ denote the solution to $\exp(1/x)=x$ and let $α>α^*$. We prove that, for any triangle-free graph $G=(V,E)$ with maximum degree $Δ$, for all $q\geqαΔ+1$, the mixing time of the Glaub…
