Let $mad(G)$ denote the maximum average degree (over all subgraphs) of $G$ and let $χ_i(G)$ denote the injective chromatic number of $G$. We prove that if $mad(G) \leq 5/2$, then $χ_i(G)\leqΔ(G) + 1$; and if $mad(G) < 42/19$, then $χ_i(G)=Δ(G)$. Suppose that $G$ is a planar graph with girth $g(G)$ and $Δ(G)\geq 4$. We prove that if $g(G)\geq 9$, then $χ_i(G)\leqΔ(G)+1$; similarly, if $g(G)\geq 13$, then $χ_i(G)=Δ(G)$.
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…
A coloring of edges of a graph $G$ is injective if for any two distinct edges $e_1$ and $e_2$, the colors of $e_1$ and $e_2$ are distinct if they are at distance $1$ in $G$ or in a common triangle. Naturally, the injective chromatic index of $G$, $χ'_{inj}(G)$, is the minimum number of colors needed for an injective edge-coloring of $G$. We study how large can be the injective chromatic index of $G$ in terms of maximum degree of $G$ when we have restrictions on girth and/or chromatic number of $G$. We also compare our bounds with analogous bounds on the strong chromatic index.
An injective edge-coloring $c$ of a graph $G$ is an edge-coloring such that if $e_1$, $e_2$, and $e_3$ are three consecutive edges in $G$ (they are consecutive if they form a path or a cycle of length three), then $e_1$ and $e_3$ receive different colors. The minimum integer $k$ such that, $G$ has an injective edge-coloring with $k$ colors, is called the injective chromatic index of $G$ ($χ'_{\textrm{inj}}(G)$). This parameter was introduced by Cardoso et \textit{al.} \cite{CCCD} motivated by the Packet Radio Network problem. They proved that computing $χ'_{\textrm{inj}}(G)$ of a graph $G$ is …
Three edges $e_{1}, e_{2}$ and $e_{3}$ in a graph $G$ are consecutive if they form a path (in this order) or a cycle of length three. An injective edge coloring of a graph $G = (V,E)$ is a coloring $c$ of the edges of $G$ such that if $e_{1}, e_{2}$ and $e_{3}$ are consecutive edges in $G$, then $c(e_{1})\neq c(e_3)$. The injective edge coloring number $χ_{i}^{'}(G)$ is the minimum number of colors permitted in such a coloring. In this paper, exact values of $χ_{i}^{'}(G)$ for several classes of graphs are obtained, upper and lower bounds for $χ_{i}^{'}(G)$ are introduced and it is proven that…
An injective coloring of a graph $G$ is an assignment of colors to the vertices of $G$ so that any two vertices with a common neighbor have distinct colors. A graph $G$ is injectively $k$-choosable if for any list assignment $L$, where $|L(v)| \geq k$ for all $v \in V(G)$, $G$ has an injective $L$-coloring. Injective colorings have applications in the theory of error-correcting codes and are closely related to other notions of colorability. In this paper, we show that subcubic planar graphs with girth at least 6 are injectively 5-choosable. This strengthens a result of Lužar, Škrekovski, and T…
An injective coloring of a graph is a vertex coloring where two vertices with common neighbor receive distinct colors. The minimum integer $k$ that $G$ has a $k-$injective coloring is called injective chromatic number of $G$ and denoted by $χ_i(G)$. In this paper, the injective chromatic number of outerplanar graphs with maximum degree $Δ$ and girth $g$ is studied. It is shown that for every outerplanar graph, $χ_i(G)\leq Δ+2$, and this bound is tight. Then, it is proved that for outerplanar graphs with $Δ=3$, $χ_i(G)\leq Δ+1$ and the bound is tight for outerplanar graphs of girth three and $4…
Let G be a simple graph. A coloring of vertices of G is called (i) a 2-proper coloring if vertices at distance 2 receive distinct colors; (ii) an injective coloring if vertices possessing a common neighbor receive distinct colors; (iii) a square coloring if vertices at distance at most 2 receive distinct colors. In this paper, we study inequalities of Nordhaus-Guddam type for the 2-proper chromatic number, the injective chromatic number, and the square chromatic number.
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…
Let $\mad(G)$ denote the maximum average degree (over all subgraphs) of $G$ and let $χ_i(G)$ denote the injective chromatic number of $G$. We prove that if $Δ\geq 4$ and $\mad(G)<\frac{14}5$, then $χ_i(G)\leqΔ+2$. When $Δ=3$, we show that $\mad(G)<\frac{36}{13}$ implies $χ_i(G)\le 5$. In contrast, we give a graph $G$ with $Δ=3$, $\mad(G)=\frac{36}{13}$, and $χ_i(G)=6$.
We consider realizations of a graph in the plane such that the distances between adjacent vertices satisfy the constraints given by an edge labeling. If there are infinitely many such realizations, counted modulo rigid motions, the labeling is called flexible. The existence of a flexible labeling, possibly non-generic, has been characterized combinatorially by the existence of a so called NAC-coloring. Nevertheless, the corresponding realizations are often non-injective. In this paper, we focus on flexible labelings with infinitely many injective realizations. We provide a necessary combinator…
DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvorak and Postle. We introduce and study $(i,j)$-defective DP-colorings of multigraphs. We concentrate on sparse multigraphs and consider $f_{DP}(i,j,n)$ --- the minimum number of edges that may have an $n$-vertex $(i,j)$-critical multigraph, that is, a multigraph $G$ that has no $(i,j)$-defective DP-coloring but whose every proper subgraph has such a coloring. For every $i$ and $j$, we find linear lower bounds on $f_{DP}(i,j,n)$ that are exact for infinitely many $n$.
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.
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…
We consider the following type of question: Given a partial proper $d$-edge coloring of the $d$-dimensional hypercube $Q_d$, and lists of allowed colors for the non-colored edges of $Q_d$,can we extend the partial coloring to a proper $d$-edge coloring using only colors from the lists? We prove that this question has a positive answer in the case when both the partial coloring and the color lists satisfy certain sparsity conditions.
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 $n$.
Let $Ω_q$ denote the set of proper $q$-colorings of the random graph $G_{n,m}, m=dn/2$ and let $H_q$ be the graph with vertex set $Ω_q$ and an edge $\{σ,τ\}$ where $σ,τ$ are mappings $[n]\to[q]$ iff $h(σ,τ)=1$. Here $h(σ,τ)$ is the Hamming distance $|\{v\in [n]:σ(v)\neqτ(v)\}|$. We show that w.h.p. $H_q$ contains a single giant component containing almost all colorings in $Ω_q$ if $d$ is sufficiently large and $q\geq \frac{cd}{\log d}$ for a constant $c>3/2$.
A \textit{star $k$-coloring} of a graph $G$ is a proper (vertex) $k$-coloring of $G$ such that the vertices on a path of length three receive at least three colors. Given a graph $G$, its \textit{star chromatic number}, denoted $χ_s(G)$, is the minimum integer $k$ for which $G$ admits a star $k$-coloring. Studying star coloring of sparse graphs is an active area of research, especially in terms of the maximum average degree of a graph; the \textit{maximum average degree}, denoted $mad(G)$, of a graph $G$ is $\max\left\{ \frac{2|E(H)|}{|V(H)|}:{H \subset G}\right\}$. It is known that for a grap…
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 $5$-…
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…
A majority coloring of a directed graph is a vertex-coloring in which every vertex has the same color as at most half of its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that every digraph has a majority 4-coloring and conjectured that every digraph admits a majority 3-coloring. We verify this conjecture for digraphs with chromatic number at most 6 or dichromatic number at most 3. We obtain analogous results for list coloring: We show that every digraph with list chromatic number at most 6 or list dichromatic number at most 3 is majority 3-choosable. We deduce that digr…
DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvořák and Postle. In this paper we introduce and study the fractional DP-chromatic number $χ_{DP}^\ast(G)$. We characterize all connected graphs $G$ such that $χ_{DP}^\ast(G) \leqslant 2$: they are precisely the graphs with no odd cycles and at most one even cycle. By a theorem of Alon, Tuza, and Voigt, the fractional list-chromatic number $χ_\ell^\ast(G)$ of any graph $G$ equals its fractional chromatic number $χ^\ast(G)$. This equality does not extend to fractional DP-colorings. Mo…
Consider a graph $G = (V, E)$ and, for each vertex $v \in V$, a subset $Σ(v)$ of neighbors of $v$. A $Σ$-coloring is a coloring of the elements of $V$ so that vertices appearing together in some $Σ(v)$ receive pairwise distinct colors. An obvious lower bound for the minimum number of colors in such a coloring is the maximum size of a set $Σ(v)$, denoted by $ρ(Σ)$. In this paper we study graph classes $F$ for which there is a function $f$, such that for any graph $G \in F$ and any $Σ$, there is a $Σ$-coloring using at most $f(ρ(Σ))$ colors. It is proved that if such a function exists for a clas…
A proper coloring of the vertices of a graph is called a \emph{star coloring} if the union of every two color classes induces a star forest. The star chromatic number $χ_s(G)$ is the smallest number of colors required to obtain a star coloring of $G$. In this paper, we study the relationship between the star chromatic number $χ_s(G)$ and the maximum average degree $\mbox{Mad}(G)$ of a graph $G$. We prove that: (1) If $G$ is a graph with $\mbox{Mad}(G) < \frac{26}{11}$, then $χ_s(G)\leq 4$. (2) If $G$ is a graph with $\mbox{Mad}(G) < \frac{18}{7}$ and girth at least 6, then $χ_s(G)\leq 5$. (3) …
An $r$-augmented tree is a rooted tree plus $r$ edges added from each leaf to ancestors. For $d,g,r\in\mathbb{N}$, we construct a bipartite $r$-augmented complete $d$-ary tree having girth at least $g$. The height of such trees must grow extremely rapidly in terms of the girth. Using the resulting graphs, we construct sparse non-$k$-choosable bipartite graphs, showing that maximum average degree at most $2(k-1)$ is a sharp sufficient condition for $k$-choosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of non-$k$-colorable graphs and hyperg…
A {\em strong $k$-edge-coloring} of a graph $G$ is a mapping from $E(G)$ to $\{1,2,\ldots,k\}$ such that every two adjacent edges or two edges adjacent to the same edge receive distinct colors. The {\em strong chromatic index} $χ_s'(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ admits a strong $k$-edge-coloring. We give bounds on $χ_s'(G)$ in terms of the maximum degree $Δ(G)$ of a graph $G$. when $G$ is sparse, namely, when $G$ is $2$-degenerate or when the maximum average degree ${\rm Mad}(G)$ is small. We prove that the strong chromatic index of each $2$-degenerate graph $G$ …
Fix a graph $G$, a list-assignment $L$ for $G$, and $L$-colorings $α$ and $β$. An $L$-recoloring sequence, starting from $α$, recolors a single vertex at each step, so that each resulting intermediate coloring is a proper $L$-coloring. An $L$-recoloring sequence transforms $α$ to $β$ if its initial coloring is $α$ and its final coloring is $β$. We prove there exists an $L$-recoloring sequence that transforms $α$ to $β$ and recolors each vertex at most a constant number of times if (i) $G$ is triangle-free and planar and $L$ is a 7-assignment, or (ii) $\mathrm{mad}(G)<17/5$ and $L$ is a 6-assig…
Fix $k \geq 3$, and let $G$ be a $k$-uniform hypergraph with maximum degree $Δ$. Suppose that for each $l = 2, ..., k-1$, every set of l vertices of G is in at most $Δ^{(k-l)/(k-1)}/f$ edges. Then the chromatic number of $G$ is $O( (Δ/\log f)^{1/(k-1)})$. This extends results of Frieze and the second author and Bennett and Bohman. A similar result is proved for 3-uniform hypergraphs where every vertex lies in few triangles. This generalizes a result of Alon, Krivelevich, and Sudakov, who proved the result for graphs. Our main new technical contribution is a deviation inequality for positive ra…
A proper vertex coloring of a graph $G$ is equitable if the sizes of color classes differ by at most one. The equitable chromatic threshold $χ_{eq}^*(G)$ of $G$ is the smallest integer $m$ such that $G$ is equitably $n$-colorable for all $n\ge m$. We show that for planar graphs $G$ with minimum degree at least two, $χ_{eq}^*(G)\le 4$ if the girth of $G$ is at least $10$, and $χ_{eq}^*(G)\le 3$ if the girth of $G$ is at least $14$.
For a graph $G$, let $G^2$ be the graph with the same vertex set as $G$ and $xy \in E(G^2)$ when $x \neq y$ and $d_G(x,y) \leq 2$. Bonamy, Lévêque, and Pinlou conjectured that if $mad (G) < 4 - \frac{2}{c+1}$ and $Δ(G)$ is large, then $χ_\ell(G^2) \leq Δ(G) + c$. We prove that if $c \geq 3$, $mad (G) < 4 - \frac{4}{c+1}$, and $Δ(G)$ is large, then $χ_\ell(G^2) \leq Δ(G) + c$. Dvořák, Krá\soft{l}, Nejedlý, and Škrekovski conjectured that $χ(G^2) \leq Δ(G) +2$ when $Δ(G)$ is large and $G$ is planar with girth at least $5$; our result implies $χ(G^2) \leq Δ(G) +6$.
List coloring $C_3$-free planar graphs with a sparse matching of restricted lists
University of Colorado Denver（丹佛大学）；University of South Carolina（南卡罗来纳大学）；University of Virginia（弗吉尼亚大学）；University of Nebraska-Lincoln（内布拉斯加大学林肯分校）；Middle Tennessee State University（中田纳西州立大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
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.
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 $2$-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.
The \emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a nonempty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. The score of the game is the sum of the sizes of all sets marked by Lister. The goal of Painter is to minimize the score, while Lister tries to maximize it. We provide strategies for Painter on various classes of graphs whose vertices can be partitioned into a bounded number of sets inducing fores…
A coloring of the edges of a graph $G$ is strong if each color class is an induced matching of $G$. The strong chromatic index of $G$, denoted by $χ_{s}^{\prime}(G)$, is the least number of colors in a strong edge coloring of $G$. In this note we prove that $χ_{s}^{\prime}(G)\leq (4k-1)Δ(G)-k(2k+1)+1$ for every $k$-degenerate graph $G$. This confirms the strong version of conjecture stated recently by Chang and Narayanan [3]. Our approach allows also to improve the upper bound from [3] for chordless graphs. We get that $% χ_{s}^{\prime}(G)\leq 4Δ-3$ for any chordless graph $G$. Both bounds rem…
On $(1,2^4)$ and $(1,2^5)$-packing edge-coloring of sparse subcubic graphs
Konkuk University（建国大学）；Korea Institute for Advanced Study (KIAS)（韩国高等研究院）；Xi’an Jiaotong-Liverpool University（西交利物浦大学）；Guizhou Minzu University（贵州民族大学）
DP-coloring (or correspondence coloring) is a generalization of list coloring that has been widely studied since its introduction by Dvořák and Postle in 2015. As the analogue of the chromatic polynomial of a graph $G$, $P(G,m)$, and the list color function, $P_{\ell}(G,m)$, the DP color function of $G$, denoted by $P_{DP}(G,m)$, counts the minimum number of DP-colorings over all possible $m$-fold covers. It follows that $P_{DP}(G,m) \le P_{\ell}(G,m) \le P(G,m)$. A function $f$ is chromatic-adherent if for every graph $G$, $f(G,a) = P(G,a)$ for some $a \geq χ(G)$ implies that $f(G,m) = P(G,m)…
