A graph is $P_t$-free if it contains no induced subgraph isomorphic to a $t$-vertex path. A graph is not bipartite if and only if it contains an induced subgraph isomorphic to a $k$-vertex cycle, where $k$ is odd. We focus on the 3-coloring problem for $P_t$-free graphs that have only one prescribed induced odd cycle length. For any integer $t$ and any odd integer $k$, let $\mathcal{G}_{t,k}$ be the class of graphs that are $P_{t}$-free and all their induced odd cycles must be $C_k$. In this paper, we present a polynomial-time algorithm that solves the 3-coloring problem for any graph in $\mathcal{G}_{10,7}$.
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…
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…
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…
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 …
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 …
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…
We show the following for every sufficiently connected graph $G$, any vertex subset $S$ of $G$, and given integer $k$: there are $k$ disjoint odd cycles in $G$ each containing a vertex of $S$ or there is set $X$ of at most $2k-2$ vertices such that $G-X$ does not contain any odd cycle that contains a vertex of $S$. We prove this via an extension of Kawarabayashi and Reed's result about parity-$k$-linked graphs (Combinatorica 29, 215-225). From this result it is easy to deduce several other well known results about the Erdős-Pósa property of odd cycles in highly connected graphs. This strengthe…
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…
It's well known that every planar graph is $4$-colorable. A toroidal graph is a graph that can be embedded on a torus. It's proved that every toroidal graph is $7$-colorable. A proper coloring of a graph is called \emph{odd} if every non-isolated vertex has at least one color that appears an odd number of times in its neighborhood. The smallest number of colors that admits an odd coloring of a graph $ G $ is denoted by $χ_{o}(G)$. In this paper, we prove that if $G$ is tortoidal, then $χ_{o}\left({G}\right)\le9$; Note that $K_7$ is a toroidal graph, the upper bound is no less than $7$.
In this paper, some results concerning the colorings of graph powers are presented. The notion of helical graphs is introduced. We show that such graphs are hom-universal with respect to high odd-girth graphs whose $(2t+1)$st power is bounded by a Kneser graph. Also, we consider the problem of existence of homomorphism to odd cycles. We prove that such homomorphism to a $(2k+1)$-cycle exists if and only if the chromatic number of the $(2k+1)$st power of $S_2(G)$ is less than or equal to 3, where $S_2(G)$ is the 2-subdivision of $G$. We also consider Nešetřil's Pentagon problem. This problem is…
We study quantum analogs of graph colorings and chromatic number. Initially defined via an interactive protocol, quantum colorings can also be viewed as a natural operator relaxation of graph coloring. Since there is no known algorithm for producing nontrivial quantum colorings, the existing examples rely on ad hoc constructions. Almost all of the known constructions of quantum $d$-colorings start from $d$-dimensional orthogonal representations. We show the limitations of this method by exhibiting, for the first time, a graph with a 3-dimensional orthogonal representation which cannot be quant…
An \emph{equitable coloring} of a graph is a proper vertex coloring such that the sizes of every two color classes differ by at most 1. Chen, Lih, and Wu conjectured that every connected graph $G$ with maximum degree $Δ\geq 2$ has an equitable coloring with $Δ$ colors, except when $G$ is a complete graph or an odd cycle or $Δ$ is odd and $G=K_{Δ,Δ}.$ Nakprasit proved the conjecture holds for planar graphs with maximum degree at least 9. Zhu and Bu proved that the conjecture holds for every $C_3$-free planar graph with maximum degree at least 8 and for every planar graph without $C_4$ and $C_5$…
In 1999, at one of his last public lectures, Tutte discussed a question he had considered since the times of the Four Color Conjecture. He asked whether the 4-coloring complex of a planar triangulation could have two components in which all colorings had the same parity. In this note we answer Tutte's question to the contrary of his speculations by showing that there are triangulations of the plane whose coloring complexes have arbitrarily many even and odd components. We end up with a closely related conjecture, which is based on an extensive computation, and which claims that for every plana…
