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.
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.
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.
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.
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…
A $2$-distance $k$-coloring of a graph is a proper $k$-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance ($Δ+2$)-coloring for graphs with maximum average degree less than $\frac{8}{3}$ (resp. $\frac{14}{5}$) and maximum degree $Δ\geq 6$ (resp. $Δ\geq 10$). As a corollary, every planar graph with girth at least $8$ (resp. $7$) and maximum degree $Δ\geq 6$ (resp. $Δ\geq 10$) admits a $2$-distance $(Δ+2)$-coloring.
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…
Let $c\geq 2$ and $p\geq c$ be two integers. We will call a proper coloring of the graph $G$ a \textit{$(c,p)$-nondegenerate}, if for any vertex of $G$ with degree at least $p$ there are at least $c$ vertices of different colors adjacent to it. In our work we prove the following result, which generalizes Brook's Theorem. Let $D\geq 3$ and $G$ be a graph without cliques on $D+1$ vertices and the degree of any vertex in this graph is not greater than $D$. Then for every integer $c\geq 2$ there is a proper $(c,p)$-nondegenerate vertex $D$-coloring of $G$, where $p=(c^3+8c^2+19c+6)(c+1).$ During t…
A $2$-distance $k$-coloring of a graph is a proper $k$-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance ($Δ+1$)-coloring for graphs with maximum average degree less than $\frac{18}{7}$ and maximum degree $Δ\geq 7$. As a corollary, every planar graph with girth at least $9$ and $Δ\geq 7$ admits a $2$-distance $(Δ+1)$-coloring. The proof uses the potential method to reduce new configurations compared to classic approaches on $2$-distance coloring.
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.
An equitable tree-$k$-coloring of a graph is a vertex coloring on $k$ colors so that every color class incudes a forest and the sizes of any two color classes differ by at most one.This kind of coloring was first introduced in 2013 and can be used to formulate the structure decomposition problem on the communication network with some security considerations. In 2015, Esperet, Lemoine and Maffray showed that every $d$-degenerate graph admits an equitable tree-$k$-coloring for every $k\geq 3^{d-1}$. Motivated by this result, we attempt to lower their exponential bound to a linear bound. Precisel…
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 2-distance list k-coloring of a graph is a proper coloring of the vertices where each vertex has a list of at least k available colors and vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance list $(Δ+ 3)$-coloring for graphs with maximum average degree less than $\frac83$ and maximum degree $Δ\geq 4$ as well as graphs with maximum average degree less than $\frac{14}5$ and maximum degree $Δ\geq 6$.
A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total coloring, the $(p,1)$-total labelling, and the equitable edge coloring of 1-planar graphs. More precisely, we verify the well-known List Edge Coloring Conjecture and List Total Coloring Conjecture for 1-planar graph with maximum degree at least 18, prove that the $(p,1)$-total labelling number of every 1-planar graph $G$ is at most $Δ(G)+2p-2$ provided that $Δ(G)\…
The reconfiguration graph $R_k(G)$ for the $k$-colorings of a graph $G$ has as vertex set the set of all possible $k$-colorings of $G$ and two colorings are adjacent if they differ in the color of exactly one vertex of $G$. Let $d, k \geq 1$ be integers such that $k \geq d+1$. We prove that for every $ε> 0$ and every graph $G$ with $n$ vertices and maximum average degree $d - ε$, $R_k(G)$ has diameter $O(n(\log n)^{d - 1})$. This significantly strengthens several existing results.
Total coloring is a variant of edge coloring where both vertices and edges are to be colored. A graph is totally $k$-choosable if for any list assignment of $k$ colors to each vertex and each edge, we can extract a proper total coloring. In this setting, a graph of maximum degree $Δ$ needs at least $Δ+1$ colors. In the planar case, Borodin proved in 1989 that $Δ+2$ colors suffice when $Δ$ is at least 9. We show that this bound also holds when $Δ$ is $8$.
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.
An equitable tree-$k$-coloring of a graph is a vertex $k$-coloring such that each color class induces a forest and the size of any two color classes differ by at most one. In this work, we show that every interval graph $G$ has an equitable tree-$k$-coloring for any integer $k\geq \lceil(Δ(G)+1)/2\rceil$, solving a conjecture of Wu, Zhang and Li (2013) for interval graphs, and furthermore, give a linear-time algorithm for determining whether a proper interval graph admits an equitable tree-$k$-coloring for a given integer $k$. For disjoint union of split graphs, or $K_{1,r}$-free interval grap…
An edge-coloring of a graph $G$ with colors $1,...,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if $G$ has an interval $t$-coloring for some positive integer $t$. Let $\mathfrak{N}$ be the set of all interval colorable graphs. For a graph $G\in \mathfrak{N}$, the least and the greatest values of $t$ for which $G$ has an interval $t$-coloring are denoted by $w(G)$ and $W(G)$, respectively. In this paper we first show that if $G$ is an $r$-regular grap…
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.
Let $G$ be a graph and $R\subseteq V(G)$. A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an $R$-sequential $t$-coloring if the edges incident to each vertex $v\in R$ are colored by the colors $1,\ldots,d_{G}(v)$, where $d_{G}(v)$ is the degree of the vertex $v$ in $G$. In this note, we show that if $G$ is a graph with $Δ(G)-δ(G)\leq 1$ and $χ^{\prime}(G)=Δ(G)=r$ ($r\geq 3$), then $G$ has an $R$-sequential $r$-coloring with $\vert R\vert \geq \left\lceil\frac{(r-1)n_{r}+n}{r}\right\rceil$, where $n=\vert V(G)\vert$ and $n_{r}=\vert\{v\in V(G):d_{G}(v)=r\}\vert$. As a c…
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…
An exact $(k,d)$-coloring of a graph $G$ is a coloring of its vertices with $k$ colors such that each vertex $v$ is adjacent to exactly $d$ vertices having the same color as $v$. The exact $d$-defective chromatic number, denoted $χ_d^=(G)$, is the minimum $k$ such that there exists an exact $(k,d)$-coloring of $G$. In an exact $(k,d)$-coloring, which for $d=0$ corresponds to a proper coloring, each color class induces a $d$-regular subgraph. We give basic properties for the parameter and determine its exact value for cycles, trees, and complete graphs. In addition, we establish bounds on $χ_d^…
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…
Let $\binom{X}{h}$ be the collection of all $h$-subsets of an $n$-set $X\supseteq Y$. Given a coloring (partition) of a set $S\subseteq \binom{X}{h}$, we are interested in finding conditions under which this coloring is extendible to a coloring of $\binom{X}{h}$ so that the number of times each element of $X$ appears in each color class (all sets of the same color) is the same number $r$. The case $S=\varnothing, r=1$ was studied by Sylvester in the 18th century, and remained open until the 1970s. The case $h=2,r=1$ is extensively studied in the literature and is closely related to completing …
For two positive integers $r$ and $s$ with $r\geq 2s-2$, if $G$ is a graph of order $3r+4s$ such that $d(x)+d(y)\geq 4r+4s$ for every $xy\not\in E(G)$, then $G$ independently contains $r$ triangles and $s$ quadrilaterals, which partially prove the El-Zahar's Conjecture.
A graph is $k$-degenerate if every subgraph has minimum degree at most $k$. We provide lower bounds on the size of a maximum induced 2-degenerate subgraph in a triangle-free planar graph. We denote the size of a maximum induced 2-degenerate subgraph of a graph $G$ by $α_2(G)$. We prove that if $G$ is a connected triangle-free planar graph with $n$ vertices and $m$ edges, then $α_2(G) \geq \frac{6n - m - 1}{5}$. By Euler's Formula, this implies $α_2(G) \geq \frac{4}{5}n$. We also prove that if $G$ is a triangle-free planar graph on $n$ vertices with at most $n_3$ vertices of degree at most thre…
A strong edge coloring of a graph is a proper edge coloring in which every color class is an induced matching. The strong chromatic index $χ_s'(G)$ of a graph $G$ is the minimum number of colors in a strong edge coloring of $G$. Let $Δ\geq 4$ be an integer. In this note, we study the odd graphs and show the existence of some special walks. By using these results and Chang's ideas in [Discuss. Math. Graph Theory 34 (4) (2014) 723--733], we show that every planar graph with maximum degree at most $Δ$ and girth at least $10 Δ- 4$ has a strong edge coloring with $2Δ- 1$ colors. In addition, we pro…
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.
We prove that for every integer $t\geq 1$, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most $t$ points is $χ$-bounded. This is essentially the strongest $χ$-boundedness result one can get for this kind of graph classes. As a corollary, we prove that for any fixed integers $k\geq 2$ and $t\geq 1$, every $k$-quasi-planar topological graph on $n$ vertices with any two edges crossing at most $t$ times has $O(n\log n)$ edges.
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.
We show that the natural Glauber dynamics mixes rapidly and generates a random proper edge-coloring of a graph with maximum degree $Δ$ whenever the number of colors is at least $q\geq (\frac{10}{3} + ε)Δ$, where $ε>0$ is arbitrary and the maximum degree satisfies $Δ\geq C$ for a constant $C = C(ε)$ depending only on $ε$. For edge-colorings, this improves upon prior work \cite{Vig99, CDMPP19} which show rapid mixing when $q\geq (\frac{11}{3}-ε_0 ) Δ$, where $ε_0 \approx 10^{-5}$ is a small fixed constant. At the heart of our proof, we establish a matrix trickle-down theorem, generalizing Oppenh…
We prove that 3-Coloring remains NP-hard on 4- and 5-regular planar Hamiltonian graphs, strengthening the results of Dailey [Disc. Math.'80] and Fleischner and Sabidussi [J. Graph. Theor.'02]. Moreover, we prove that 3-Coloring remains NP-hard on $p$-regular Hamiltonian graphs for every $p\geq 6$ and $p$-ordered regular Hamiltonian graphs for every $p\geq 3$.
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…
For integers r and k > 0(k>r),a conditional (k, r)-coloring of a graph G is a proper k-coloring of G such that every vertex v of G has at least min{r,d(v)} differently colored neighbors, where d(v) is the degree of v. In this note, for different values of r we obtain the conditional chromatic number of a grid $G(2,n) \cong P_2 \ \Box \ P_n$, $C_n^2$ and the strong product of $P_n$ and $P_m$ (n,m being positive integers). Also, for integers $n \geq 3$ and $t \geq 1$ the second order conditional chromatic number (also known as dynamic chromatic number) of the (t,n)-web graph is obtained.
We consider the k-strong conflict-free coloring of a set of points on a line with respect to a family of intervals: Each point on the line must be assigned a color so that the coloring has to be conflict-free, in the sense that in every interval I there are at least k colors each appearing exactly once in I. In this paper, we present a polynomial algorithm for the general problem; the algorithm has an approximation factor 5-2/k when k\geq2 and approximation factor 2 for k=1. In the special case the family contains all the possible intervals on the given set of points, we show that a 2 approxim…
