We improve upper bounds on the chromatic number proven independently in \cite{reedNote} and \cite{ingo}. Our main lemma gives a sufficient condition for two paths in graph to be completely joined. Using this, we prove that if a graph has an optimal coloring with more than $\fracω{2}$ singleton color classes, then it satisfies $χ\leq \frac{ω+ Δ+ 1}{2}$. It follows that a graph satisfying $n - Δ< α+ \frac{ω- 1}{2}$ must also satisfy $χ\leq \frac{ω+ Δ+ 1}{2}$, improving the bounds in \cite{reedNote} and \cite{ingo}. We then give a simple argument showing that if a graph satisfies $χ> \frac{n + 3 - α}{2}$, then it also satisfies $χ(G) \leq \left\lceil\frac{ω(G) + Δ(G) + 1}{2}\right\rceil$. From this it follows that a graph satisfying $n - Δ< α+ ω$ also satisfies $χ(G) \leq \left\lceil\frac{ω(G) + Δ(G) + 1}{2}\right\rceil$ improving the bounds in \cite{reedNote} and \cite{ingo} even further at the cost of a ceiling. In the next sections, we generalize our main lemma to constrained colorings (e.g. r-bounded colorings). We present a generalization of Reed's conjecture to r-bounded colorings and prove the conjecture for graphs with maximal degree close to their order. Finally, we outline some applications (in \cite{BorodinKostochka} and \cite{ColoringWithDoublyCriticalEdge}) of the theory presented here to the Borodin-Kostochka conjecture and coloring graphs containing a doubly critical edge.
A planar graph can be embedded in a piecewise linear manifold, and the lattice on each linear piece can be colored with 3-coloring. If a planar graph can be colored with multiple 3-coloring, i.e. coloring the graph in pieces with different 3-color subsets of 4 colors, then the graph is 4-colorable. In this paper, multiple 3-coloring was introduced, and then the combination and partition of planar graphs for multiple 3-coloring was studied. The study reveals that planar graphs can generally be decomposed into independent subgraphs, and each subgraph can be triangulated into a symmetric structur…
We consider the graph coloring game, a game in which two players take turns properly coloring the vertices of a graph, with one player attempting to complete a proper coloring, and the other player attempting to prevent a proper coloring. We show that if a graph $G$ has a proper coloring in which the game coloring number of each bicolored subgraph is bounded, then the game chromatic number of $G$ is bounded. As a corollary to this result, we show that for two graphs $G_1$ and $G_2$ with bounded game coloring number, the Cartesian product $G_1 \square G_2$ has bounded game chromatic number, ans…
Given a 3-colorable graph $X$, the 3-coloring complex $B(X)$ is the graph whose vertices are all the independent sets which occur as color classes in some 3-coloring of $X$. Two color classes $C,D \in V(B(X))$ are joined by an edge if $C$ and $D$ appear together in a 3-coloring of $X$. The graph $B(X)$ is 3-colorable. Graphs for which $B(B(X))$ is isomorphic to $X$ are termed reflexive graphs. In this paper, we consider 3-edge-colorings of cubic graphs for which we allow half-edges. Then we consider the 3-coloring complexes of their line graphs. The main result of the paper is a surprising out…
A fall coloring of a graph G is a proper coloring of G with k colors such that each vertex sees all k colors on its closed neighborhood. In this short note, we characterize all fall colorings of Kneser graphs of type KG(n,2).
We define a method for edge coloring signed graphs and what it means for such a coloring to be proper. Our method has many desirable properties: it specializes to the usual notion of edge coloring when the signed graph is all-negative, it has a natural definition in terms of vertex coloring of a line graph, and the minimum number of colors required for a proper coloring of a signed simple graph is bounded above by Δ + 1 in parallel with Vizing's Theorem. In fact, Vizing's Theorem is a special case of the more difficult theorem concerning signed graphs.
A total dominator coloring of a graph G is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a total dominator coloring. In this article, we study the total dominator coloring on middle graphs by giving several bounds for the case of general graphs and trees. Moreover, we calculate explicitely the total dominator chromatic number of the middle graph of several known families of graphs.
The idea that those different from you are "unfriendly" is captured in the definition of unfriendly 2-colorings in graph theory in a paper by Aharoni, Milner and Prikry, where they prove that every finite graph has an unfriendly coloring. We give a more general definition for all n>1, that we call "integrated" rather than "unfriendly." Then we prove that every finite graph has an integrated n-coloring, n>1. We then give some applications to various graph coloring problems and to some max-cut problems.
Let $G$ be a simple graph of order $n$. A majority dominator coloring of a graph $G$ is proper coloring in which each vertex of the graph dominates at least half of one color class. The majority dominator chromatic number $χ_{md}(G)$ is the minimum number of color classes in a majority dominator coloring of $G$. In this paper we study properties of the majority dominator coloring of a graph. We obtain tight upper and lower bounds in terms of chromatic number, dominator chromatic number, maximum degree, domination and independence number. We also study majority dominator coloring number of sele…
In this research, the exact results on $r$-dynamic coloring of lexicographic product of path with star graph, path with double star graph, path with triple star graph and finally complete graph with path are obtained.
A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvořák and Postle. Essentially, DP-coloring assigns an arbitrary matching between lists of colors at adjacent vertices, as opposed to only matching identical colors as is done for list-coloring. Several results on list-coloring of planar graphs have since been extended to the setting of DP-coloring. We note that list-coloring results do not always extend to DP-coloring results. Our main result in this paper is to prove that every planar graph without cycles of length $\{4, a, b, 9\}$ for $a, b \in \{6, 7, 8…
A total dominator coloring of a graph $G$ is a proper coloring of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a total dominator coloring of it. Here, we study the total dominator coloring on central graphs by giving some tight bounds for the total dominator chromatic number of the central of a graph, join of two graphs and Nordhaus-Gaddum-like relations. Also we will calculate the total dominator chromatic number of the central of a path, a cycle, a wheel, a comp…
An $\ell$-facial edge-coloring of a plane graph is a coloring of its edges such that any two edges at distance at most $\ell$ on a boundary walk of any face receive distinct colors. It is the edge-coloring variant of the $\ell$-facial vertex coloring, which arose as a generalization of the well-known cyclic coloring. It is conjectured that at most $3\ell + 1$ colors suffice for an $\ell$-facial edge-coloring of any plane graph. The conjecture has only been confirmed for $\ell \le 2$, and in this paper, we prove its validity for $\ell = 3$.
For a graph $G$ of order $n$ a maximal edge coloring is a proper edge coloring with $χ'(K_n)$ colors such that adding any edge to $G$ in any color makes it improper. Meszka and Tyniec proved that for some values of the number of edges there are no graphs with a maximal edge coloring, while for some other values, they provided constructions of such graphs. However, for many values of the number of edges determining whether there exists any graph with a maximal edge coloring remained open. We give a complete solution of this problem.
A proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a $k$-regular graph at least $2k-1$ colors are needed. We show that a $k$-regular graph admits a strong edge coloring with $2k-1$ colors if and only if it covers the Kneser graph $K(2k-1,k-1)$. In particular, a cubic graph is strongly $5$-edge-colorable whenever it covers the Petersen graph. One of the implications of this result is that a conjecture about strong edge colorings of subcubic graphs proposed by Faudree et al. [Ars Combin. 29 B (1990), …
In the past various distance based colorings on planar graphs were introduced. We turn our focus to three of them, namely $2$-distance coloring, injective coloring, and exact square coloring. A $2$-distance coloring is a proper coloring of the vertices in which no two vertices at distance $2$ receive the same color, an injective coloring is a coloring of the vertices in which no two vertices with a common neighbor receive the same color, and an exact square coloring is a coloring of the vertices in which no two vertices at distance exactly $2$ receive the same color. We prove that planar graph…
The chromatic number of signed graphs is defined recently. The coloring and clique problem of interval graphs has been studied and polynomial time algorithms are established. Here we consider these problems for signed interval graphs and prove that the coloring problem of signed interval graphs is NP-complete whereas their ordinary clique problem is in P. We also study the complexity of further related problems.
We propose the notion of a majority $k$-edge-coloring of a graph $G$, which is an edge-coloring of $G$ with $k$ colors such that, for every vertex $u$ of $G$, at most half the edges of $G$ incident with $u$ have the same color. We show the best possible results that every graph of minimum degree at least $2$ has a majority $4$-edge-coloring, and that every graph of minimum degree at least $4$ has a majority $3$-edge-coloring. Furthermore, we discuss a natural variation of majority edge-colorings and some related open problems.
An {\it odd $c$-coloring} of a graph is a proper $c$-coloring such that each non-isolated vertex has a color appearing an odd number of times on its neighborhood. This concept was introduced very recently by Petru\v sevski and \v Skrekovski and has attracted considerable attention. Cranston investigated odd colorings of graphs with bounded maximum average degree, and conjectured that every graph $G$ with $mad(G)\leq \frac{4c-4}{c+1}$ has an odd $c$-coloring for $c\geq 4$, and proved the conjecture for $c\in\{5, 6\}$. In particular, planar graphs with girth at least $7$ and $6$ have an odd $5$-…
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.
The concept of DP-coloring of a graph is a generalization of list coloring introduced by Dvořák and Postle in 2015. Multiple DP-coloring of graphs, as a generalization of multiple list coloring, was first studied by Bernshteyn, Kostochka and Zhu in 2019. This paper proves that planar graphs without 3-cycles and normally adjacent 4-cycles are $(7m, 2m)$-DP-colorable for every integer $m$. As a consequence, the strong fractional choice number of any planar graph without 3-cycles and normally adjacent 4-cycles is at most $7/2$.
DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvořák and Postle. We introduce and study $(i,j)$-defective DP-colorings of simple graphs. Let $g_{DP}(i,j,n)$ be the minimum number of edges in an $n$-vertex DP-$(i,j)$-critical graph. In this paper we determine sharp bound on $g_{DP}(i,j,n)$ for each $i\geq3$ and $j\geq 2i+1$ for infinitely many $n$.
We consider the problem of existence of perfect $2$-colorings (equitable $2$-partitions) of Hamming graphs with given parameters. We start with conditions on parameters of graphs and colorings that are necessary for their existence. Next we observe known constructions of perfect colorings and propose some new ones giving new parameters. At last, we deduce which parameters of colorings are covered by these constructions and give tables of admissible parameters of $2$-colorings in Hamming graphs $H(n,q)$ for small $n$ and $q$. Using the connection with perfect colorings, we construct an orthogon…
Given a partial edge coloring of a complete graph $K_n$ and lists of allowed colors for the non-colored edges of $K_n$, can we extend the partial edge coloring to a proper edge coloring of $K_n$ using only colors from the lists? We prove that this question has a positive answer in the case when both the partial edge coloring and the color lists satisfy certain sparsity conditions.
DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced by Dvourák and Postle (2017). Recently, Huang et al. [https://doi.org/10.1016/j.amc.2019.124562] showed that planar graphs with diameter at most two are $4$-choosable. In this paper, we will prove that planar graphs with diameter at most two are DP-$4$-colorable, which is an extension of the above result.
A {\it proper conflict-free $c$-coloring} of a graph is a proper $c$-coloring such that each non-isolated vertex has a color appearing exactly once on its neighborhood. This notion was formally introduced by Fabrici et al., who proved that planar graphs have a proper conflict-free 8-coloring and constructed a planar graph with no proper conflict-free 5-coloring. Caro, Petruševski, and Škrekovski investigated this coloring concept further, and in particular studied upper bounds on the maximum average degree that guarantees a proper conflict-free $c$-coloring for $c\in\{4,5,6\}$. Along these lin…
