图的一个正常染色被称为奇染色，如果每个非孤立顶点在其邻域中都有某种颜色出现奇数次。允许图 G 进行奇染色所需的最少颜色数记为 χ_o(G)。这一概念由 Petruševski 和 Škrekovski 提出，他们证明了如果 G 是平面图，则 χ_o(G)≤9；他们还猜想 χ_o(G)≤5。对于正实数 α，我们考虑在所有最大平均度小于 α 的图 G 中 χ_o(G) 的最大值，并将该值记为 χ_o(G_α)。我们指出，对于所有 α≥4，χ_o(G_α) 均未定义。相比之下，对于每个 α∈[0,4)，我们给出了 χ_o(G_α) 的一个（近乎紧的）上界。最后，我们证明 χ_o(G_{20/7})=5 且 χ_o(G_3)=6。这两个结果都是紧的。
英文摘要：
A proper coloring of a graph is called \emph{odd} if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. The smallest number of colors that admits an odd coloring of a graph $G$ is denoted $χ_o(G)$. This notion was introduced by Petruševski and Škrekovski, who proved that if $G$ is planar then $χ_o(G)\le 9$; they also conjectured that $χ_o(G)\le 5$. For a positive real number $α$, we consider the maximum value of $χ_o(G)$ over all graphs $G$ with maximum average degree less than $α$; we denote this value by $χ_o(\mathcal{G}_α)$. We note that $χ_o(\mathcal{G}_α)$ is undefined for all $α\ge 4$. In contrast, for each $α\in[0,4)$, we give a (nearly sharp) upper bound on $χ_o(\mathcal{G}_α)$. Finally, we prove $χ_o(\mathcal{G}_{20/7})= 5$ and $χ_o(\mathcal{G}_3)= 6$. Both of these results are sharp.
A proper vertex coloring $φ$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $φ^{-1}(c)\cap N(x)$ is odd-sized. The minimum number of colors in any odd coloring of $G$, denoted $χ_o(G)$, is the odd chromatic number. Odd colorings were recently introduced in [M.~Petruševski, R.~Škrekovski: \textit{Colorings with neighborhood parity condition}]. Here we discuss various basic properties of this new graph parameter, characterize acyclic graphs and hypercubes in terms of odd chromatic number, establish several upper bounds in regard to d…
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 $5$-…
We give an upper bound for the online chromatic number of graphs with high girth and for graphs with high oddgirth generalizing Kier- stead's algorithm for graphs that contain neither a C3 or C5 as an induced subgraph.
In this short paper, we introduce a new vertex coloring whose motivation comes from our series on odd edge-colorings of graphs. A proper vertex coloring $φ$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $φ^{-1}(c)\cap N(x)$ is odd-sized. We prove that every simple planar graph admits an odd $9$-coloring, and conjecture that $5$ colors always suffice.
A vertex coloring of a given simple graph $G=(V,E)$ with $k$ colors ($k$-coloring) is a map from its vertex set to the set of integers $\{1,2,3,\dots, k\}$. A coloring is called perfect if the multiset of colors appearing on the neighbours of any vertex depends only on the color of the vertex. We consider perfect colorings of Cayley graphs of the additive group of integers with generating set $\{1,-1,3,-3,5,-5,\dots, 2n-1,1-2n\}$ for a positive integer $n$. We enumerate perfect $2$-colorings of the graphs under consideration and state the conjecture generalizing the main result to an arbitrary…
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.
An odd coloring of a graph $G$ is a proper coloring such that any non-isolated vertex in $G$ has a coloring appears odd times on its neighbors. The odd chromatic number, denoted by $χ_o(G)$, is the minimum number of colors that admits an odd coloring of $G$. Petruševski and Škrekovski in 2021 introduced this notion and proved that if $G$ is planar, then $χ_o(G)\le9$ and conjectured that $χ_o(G)\le5$. More recently, Petr and Portier improved $9$ to $8$. A graph is $1$-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. Cranston, Lafferty and Song showe…
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 $n$.
An odd graph is a finite graph all of whose vertices have odd degrees. Given graph $G$ is decomposable into $k$ odd subgraphs if its edge set can be partitioned into $k$ subsets each of which induces an odd subgraph of $G$. The minimum value of $k$ for which such a decomposition of $G$ exists is the odd chromatic index, $χ_{o}'(G)$, introduced by Pyber (1991). For every $k\geqχ_{o}'(G)$, the graph $G$ is said to be odd $k$-edge-colorable. Apart from two particular exceptions, which are respectively odd $5$- and odd $6$-edge-colorable, the rest of connected loopless graphs are odd $4$-edge-colo…
We study the following Steinberg-type problem on circular coloring: for an odd integer $k\ge 3$, what is the smallest number $f(k)$ such that every planar graph of girth $k$ without cycles of length from $k+1$ to $f(k)$ admits a homomorphism to the odd cycle $C_k$ (or equivalently, is circular $(k,\frac{k-1}{2})$-colorable). Known results and counterexamples on Steinberg's Conjecture indicate that $f(3)\in\{6,7\}$. In this paper, we show that $f(k)$ exists if and only if $k$ is an odd prime. Moreover, we prove that for any prime $p\ge 5$, $$p^2-\frac{5}{2}p+\frac{3}{2}\le f(p)\le 2p^2+2p-5.$$ …
Strengthening Hadwiger's conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$-minor is properly $(t-1)$-colorable, this is known as the Odd Hadwiger's conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd $K_t$-minor admits a vertex $(2t-2)$-coloring such that all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$. The bound on the number of colors is optimal up to a factor of $2$, improves previous bounds for the same problem by Kawarabayashi (2008), Kang and Oum (2019), Liu and Wood (2…
In this paper we determine the chromatic number of graphs with two odd cycle lengths. Let $G$ be a graph and $L(G)$ be the set of all odd cycle lengths of $G$. We prove that: (1) If $L(G)=\{3,3+2l\}$, where $l\geq 2$, then $χ(G)=\max\{3,ω(G)\}$; (2) If $L(G)=\{k,k+2l\}$, where $k\geq 5$ and $l\geq 1$, then $χ(G)=3$. These, together with the case $L(G)=\{3,5\}$ solved in \cite{W}, give a complete solution to the general problem addressed in \cite{W,CS,KRS}. Our results also improve a classical theorem of Gyárfás which asserts that $χ(G)\le 2|L(G)|+2$ for any graph $G$.
As a strengthening of Hadwiger's conjecture, Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. We prove two weaker variants of this conjecture. Firstly, we show that for each $t \geq 2$, every graph with no odd $K_t$ minor has a partition of its vertex set into $6t-9$ sets $V_1, \dots, V_{6t-9}$ such that each $V_i$ induces a subgraph of bounded maximum degree. Secondly, we prove that for each $t \geq 2$, every graph with no odd $K_t$ minor has a partition of its vertex set into $10t-13$ sets $V_1, \dots, V_{10t-13}$ such that each $V_i$ induces a s…
We give a very short self-contained combinatorial proof of the Babson-Kozlov conjecture, by presenting a cochain whose coboundary is the desired power of the characteristic class.
For building up twin-graphic lattices towards topological cryptograph, we define four kinds of new odd-magic-type colorings: odd-edge graceful-difference total coloring, odd-edge edge-difference total coloring, odd-edge edge-magic total coloring, and odd-edge felicitous-difference total coloring in this article. Our RANDOMLY-LEAF-ADDING algorithms are based on adding randomly leaves to graphs for producing continuously graphs admitting our new odd-magic-type colorings. We use complex graphs to make caterpillar-graphic lattices and complementary graphic lattices, such that each graph in these n…
A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses at most $k$ colors. For an integer $k\geq 1$, the Gallai-Ramsey number $GR_k(H)$ of a given graph $H$ is the least positive integer $N$ such that every Gallai $k$-coloring of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $C_m$ denote the cycle on $m\ge4$ vertices and let $Θ_m$ denote the family of graphs obtained from $C_m$ by adding an additional edge joining two non-consecutive vertices. We prove that $GR_k(Θ_{2n+1})=n\cdot …
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 tree-coloring of a maximal planar graph is a proper vertex $4$-coloring such that every bichromatic subgraph, induced by this coloring, is a tree. A maximal planar graph $G$ is tree-colorable if $G$ has a tree-coloring. In this article, we prove that a tree-colorable maximal planar graph $G$ with $δ(G)\geq 4$ contains at least four odd-vertices. Moreover, for a tree-colorable maximal planar graph of minimum degree 4 that contains exactly four odd-vertices, we show that the subgraph induced by its four odd-vertices is not a claw and contains no triangles.
Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors $k$ that can ensure an odd coloring of a graph $G$ is denoted by $χ_o(G)$. We say $G$ is odd $k$-colorable if $χ_o(G)\le k$. This notion is introduced very recently by Petruševski and Škrekovski, who proved that if $G$ is planar then $ χ_{o}(G) \leq 9 $. A toroidal graph is a graph that can be embedded on a torus. Note that a $K_7$ is a toroidal graph, $χ_{o}(G)\leq7$. Tian and Yin proved that every …
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…
Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors $k$ that can ensure an odd coloring of a graph $G$ is denoted by $χ_o(G)$. We say $G$ is $k$-colorable if $χ_o(G)\le k$. This notion is introduced very recently by Petruševski and Škrekovski, who proved that if $G$ is planar then $ χ_{o}(G) \leq 9 $. A toroidal graph is a graph that can be embedded on a torus. Note that a $K_7$ is a toroidal graph, $χ_{o}(G)\leq7$. In this paper, we proved that, ever…
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…
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 …
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 $χ_i(G)=Δ(G)$.
We classify the trees on $n$ vertices with the maximum and the minimum number of certain generalized colorings, including conflict-free, odd, non-monochromatic, star, and star rainbow vertex colorings. We also extend a result of Cutler and Radcliffe on the maximum and minimum number of existence homomorphisms from a tree to a completely looped graph on $q$ vertices.
