The problem of finding the minimum number of colors to color a graph properly without containing any bicolored copy of a fixed family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of graphs (Grünbaum, 1973) where bicolored copies of $P_4$ and cycles are not allowed, respectively. In this paper, we introduce a variation of these problems and study proper coloring of graphs not containing a bicolored path of a fixed length and provide general bounds for all graphs. A $P_k$-coloring of an undirected graph $G$ is a proper vertex coloring of $G$ such that there is no bicolored copy of $P_k$ in $G,$ and the minimum number of colors needed for a $P_k$-coloring of $G$ is called the $P_k$-chromatic number of $G,$ denoted by $s_k(G).$ We provide bounds on $s_k(G)$ for all graphs, in particular, proving that for any graph $G$ with maximum degree $d\geq 2,$ and $k\geq4,$ $s_k(G)=O(d^{\frac{k-1}{k-2}}).$ Moreover, we find the exact values for the $P_k$-chromatic number of the products of some cycles and paths for $k=5,6.$
We show that for any fixed integer $m \geq 1$, a graph of maximum degree $Δ$ has a coloring with $O(Δ^{(m+1)/m})$ colors in which every connected bicolored subgraph contains at most $m$ edges. This result unifies previously known upper bounds on the number of colors sufficient for certain types of graph colorings, including star colorings, for which $O(Δ^{3/2})$ colors suffice, and acyclic colorings, for which $O(Δ^{4/3})$ colors suffice. Our proof uses a probabilistic method of Alon, McDiarmid, and Reed. This result also gives previously unknown upper bounds, including the fact that a graph o…
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…
The chromatic number of a graph is the minimum $k$ such that the graph has a proper $k$-coloring. There are many coloring parameters in the literature that are proper colorings that also forbid bicolored subgraphs. Some examples are $2$-distance coloring, acyclic coloring, and star coloring, which forbid a bicolored path on three vertices, bicolored cycles, and a bicolored path on four vertices, respectively. This notion was first suggested by Grünbaum in 1973, but no specific name was given. We revive this notion by defining an $H$-avoiding $k$-coloring to be a proper $k$-coloring that forbid…
A proper vertex coloring of a graph $G$ is called a star coloring if every two color classes induce a forest whose each component is a star, which means there is no bicolored $P_4$ in $G$. In this paper, we show that the Cartesian product of any two cycles, except $C_3 \square C_3$ and $C_3 \square C_5$, has a $5$-star coloring.
The star chromatic index of a multigraph $G$, denoted $χ'_{st}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bicolored. We survey the results of determining the star chromatic index, present the interesting proofs and techniques, and collect many open problems and conjectures.
Consider the collection of edge bicolorings of a graph that is cellularly embedded on an orientable surface. In this work, we count the number of equivalence classes of such colorings under two relations: reversing colors around a face and reversing colors around a vertex. In the case of the plane, this is well studied, but for other surfaces, the computation is more subtle. While this question can be stated purely graph theoretically, it has interesting applications in knot theory.
The star chromatic index of a multigraph $G$, denoted by $χ_{\mathrm{star}}'(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length $4$ is bicolored. In this paper, we study the star edge-coloring of Halin graphs, $k$-power graphs and the generalized Petersen graphs $P(3n, n)$.
A balanced edge-coloring of the complete graph is an edge-coloring such that every vertex is incident to each color the same number of times. In this short note, we present a construction of a balanced edge-coloring with six colors of the complete graph on $n=13^k$ vertices, for every positive integer $k$, with no rainbow $K_4$. This solves a problem by Erdős and Tuza.
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 $n \geq 3$.
We consider the problem of extending and avoiding partial edge colorings of hypercubes; that is, given a partial edge coloring $φ$ of the $d$-dimensional hypercube $Q_d$, we are interested in whether there is a proper $d$-edge coloring of $Q_d$ that agrees with the coloring $φ$ on every edge that is colored under $φ$; or, similarly, if there is a proper $d$-edge coloring that disagrees with $φ$ on every edge that is colored under $φ$. In particular, we prove that for any $d\geq 1$, if $φ$ is a partial $d$-edge coloring of $Q_d$, then $φ$ is avoidable if every color appears on at most $d/8$ edg…
For a fixed simple digraph $F$ and a given simple digraph $D$, an $F$-free $k$-coloring of $D$ is a vertex-coloring in which no induced copy of $F$ in $D$ is monochromatic. We study the complexity of deciding for fixed $F$ and $k$ whether a given simple digraph admits an $F$-free $k$-coloring. Our main focus is on the restriction of the problem to planar input digraphs, where it is only interesting to study the cases $k \in \{2,3\}$. From known results it follows that for every fixed digraph $F$ whose underlying graph is not a forest, every planar digraph $D$ admits an $F$-free $2$-coloring, a…
Let $F(n,k)$ ($f(n,k)$) denote the maximum possible size of the smallest color class in a (partial) $k$-coloring of the Boolean lattice $B_n$ that does not admit a rainbow antichain of size $k$. The value of $F(n,3)$ and $f(n,2)$ has been recently determined exactly. We prove that for any fixed $k$ if $n$ is large enough, then $F(n,k),f(n,k)=2^{(1/2+o(1))n}$ holds. We also introduce the general functions for any poset $P$ and integer $c\ge |P|$: let $F(n,c,P)$ ($f(n,c,P)$) denote the the maximum possible size of the smallest color class in a (partial) $c$-coloring of the Boolean lattice $B_n$ …
For a fixed graph $H$ on $k$ vertices, and a graph $G$ on at least $k$ vertices, we write $G\rightarrow H$ if in any vertex-coloring of $G$ with $k$ colors, there is an induced subgraph isomorphic to $H$ whose vertices have distinct colors. In other words, if $G\rightarrow H$ then a totally multicolored induced copy of $H$ is unavoidable in any vertex-coloring of $G$ with $k$ colors. In this paper, we show that, with a few notable exceptions, for any graph $H$ on $k$ vertices and for any graph $G$ which is not isomorphic to $H$, $G\not\!\rightarrow H$. We explicitly describe all exceptional ca…
Star Coloring on Some Subclasses of Chordal Graphs
Facultad de Ciencias, Universidad Nacional Autónoma de México（墨西哥国立自治大学理学院）；Instituto de Matemáticas, Universidad Nacional Autónoma de México（墨西哥国立自治大学数学研究所）；Departamento de Computação, Universidade Federal do Ceará（巴西联邦塞阿拉大学计算机系）
Given a graph $F$ and an integer $r \ge 2$, a partition $\widehat{F}$ of the edge set of $F$ into at most $r$ classes, and a graph $G$, define $c_{r, \widehat{F}}(G)$ as the number of $r$-colorings of the edges of $G$ that do not contain a copy of $F$ such that the edge partition induced by the coloring is isomorphic to the one of $F$. We think of $\widehat{F}$ as the pattern of coloring that should be avoided. The main question is, for a large enough $n$, to find the (extremal) graph $G$ on $n$ vertices which maximizes $c_{r, \widehat{F}}(G)$. This problem generalizes a question of Erd{\H o}s…
We give a simple short proof of Brooks' theorem using only induction and greedy coloring, while avoiding issues of graph connectivity. The argument generalizes easily to some extensions of Brooks' theorem, including its variants for list coloring, signed graphs coloring and correspondence coloring.
We discuss a variant of the Ramsey and the directed Ramsey problem. First, consider a complete graph on $n$ vertices and a two-coloring of the edges such that every edge is colored with at least one color and the number of bicolored edges $|E_{RB}|$ is given. The aim is to find the maximal size $f$ of a monochromatic clique which is guaranteed by such a coloring. Analogously, in the second problem we consider semicomplete digraph on $n$ vertices such that the number of bi-oriented edges $|E_{bi}|$ is given. The aim is to bound the size $F$ of the maximal transitive subtournament that is guaran…
Maximal planar graph refers to the planar graph with the most edges, which means no more edges can be added so that the resulting graph is still planar. The Four-Color Conjecture says that every planar graph without loops is 4-colorable. Indeed, in order to prove Four-Color Conjecture, it clearly suffices to show that all maximal planar graphs are 4-colorable. Since this conjecture was proposed in 1852, no mathematical proofs have been invented up until now. Maybe the main reasons lie in the following three aspects in terms of maximal planar graphs: not clearing up the structures, not figuring…
We find families of graphs $G$ and subgraphs $H$ of $G$ such that the number of edge colorings of $G$ avoiding a monochromatic coloring of $H$ is determined by lattice point counts or a Hodge structure on the cohomology of a certain toric variety. In general, this gives a class of ``anti-Ramsey theory problems'' with a geometric structure. For example, we find one for Ramsey numbers of classes of such graphs. The key observation is that our previous result expressing simplicial chromatic polynomials in terms of $h$-vectors of auxiliary simplicial complexes can be reinterpreted as one on edge c…
For $S$ a set of positive integers, and $k$ and $r$ fixed positive integers, denote by $f(S,k;r)$ the least positive integer $n$ (if it exists) such that within every $r$-coloring of $\{1,2,...,n\}$ there must be a monochromatic sequence $\{x_{1},x_{2},...,x_{k}\}$ with $x_{i}-x_{i-1} \in S$ for $2 \leq i \leq k$. We consider the existence of $f(S,k;r)$ for various choices of $S$, as well as upper and lower bounds on this function. In particular, we show that this function exists for all $k$ if $S$ is an odd translate of the set of primes and $r=2$.
A vertex coloring $ϕ$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$ either $ϕ$ uses more than $p$ colors on $H$ or there is a color that appears exactly once on $H$. Centered colorings form one of the families of parameters that allow to capture notions of sparsity of graphs: A class of graphs has bounded expansion if and only if there is a function $f$ such that for every $p\geq1$, every graph in the class admits a $p$-centered coloring using at most $f(p)$ colors. In this paper, we give upper bounds for the maximum number of colors needed in a $p$-centered colorin…
Given a line arrangement $\cal A$ with $n$ lines, we show that there exists a path of length $n^2/3 - O(n)$ in the dual graph of $\cal A$ formed by its faces. This bound is tight up to lower order terms. For the bicolored version, we describe an example of a line arrangement with $3k$ blue and $2k$ red lines with no alternating path longer than $14k$. Further, we show that any line arrangement with $n$ lines has a coloring such that it has an alternating path of length $Ω(n^2/ \log n)$. Our results also hold for pseudoline arrangements.
As an extension of the Brooks theorem, Catlin in 1979 showed that if $H$ is neither an odd cycle nor a complete graph with maximum degree $Δ(H)$, then $H$ has a vertex $Δ(H)$-coloring such that one of the color classes is a maximum independent set. Let $G$ be a connected graph of order at least $2$. A $G$-free $k$-coloring of a graph $H$ is a partition of the vertex set of $H$ into $V_1,\ldots,V_k$ such that $H[V_i]$, the subgraph induced on $V_i$, does not contain any subgraph isomorphic to $G$. As a generalization of Catlin's theorem we show that a graph $H$ has a $G$-free $\lceil{Δ(H)\over …
Given an equation, the integers $[n] = \{1, 2, \dots, n\}$ as inputs, and the colors red and blue, how can we color $[n]$ in order to minimize the number of monochromatic solutions to the equation, and what is the minimum? The answer is only known for a handful of equations, but much progress has been made on improving upper and lower bounds on minima for various equations. A well-studied characteristic an equation, which has its roots in graph Ramsey theory, is to determine if the minimum number of monochromatic solutions can be achieved (asymptotically) by uniformly random colorings. Such eq…
Ramsey Theory deals with avoiding certain patterns. When constructing an instance that avoids one pattern, it is observed that other patterns emerge. For example, repetition emerges when avoiding arithmetic progression (Van der Waerden numbers), while reflection emerges when avoiding monochromatic solutions of $a+b=c$ (Schur numbers). We exploit observed patterns when coloring a grid while avoiding monochromatic rectangles. Like many problems in Ramsey Theory, this problem has a rapidly growing search space that makes computer search difficult. Steinbach et al. obtained a solution of an 18 by …
Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?
Department for AI in Science, Society, and Technology, Zuse Institute Berlin, Germany（人工智能科学、社会与技术部门，柏林Zuse研究所）；Institute of Mathematics, Technische Universität Berlin, Germany（数学研究所，柏林技术大学）
For given graphs $G_1,\ldots,G_k$, the size-Ramsey number $\hat{R}(G_1,\ldots,G_k)$ is the smallest integer $m$ for which there exists a graph $H$ on $m$ edges such that in every $k$-edge coloring of $H$ with colors $1,\ldots,k$, $ H $ contains a monochromatic copy of $G_i$ of color $i$ for some $1\leq i\leq k$. We denote $\hat{R}(G_1,\ldots,G_k)$ by $\hat{R}_{k}(G)$ when $G_1=\cdots=G_k=G$. Haxell, Kohayakawa and Łuczak showed that the size Ramsey number of a cycle $C_n$ is linear in $n$ i.e. $\hat{R}_{k}(C_{n})\leq c_k n$ for some constant $c_k$. Their proof, is based on the regularity lemma…
The Zarankiewicz number $z(m,n;s,t)$ is the maximum number of edges in a subgraph of $K_{m,n}$ that does not contain $K_{s,t}$ as a subgraph. The bipartite Ramsey number $b(n_1, \cdots, n_k)$ is the least positive integer $b$ such that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a monochromatic copy of $K_{n_i,n_i}$ in the $i$-th color, for some $i$, $1 \le i \le k$. If $n_i=m$ for all $i$, then we denote this number by $b_k(m)$. In this paper we obtain the exact values of some Zarankiewicz numbers for quadrilateral ($s=t=2$), and we derive new bounds for diagonal mul…
