设 $ϕ_c(G)$ 为无桥图 $G$ 的循环流数。在 [Edge-colorings and circular flow numbers of regular graphs, J. Graph Theory 79 (2015) 1-7] 中已证明：对于每个 $t \geq 1$，$G$ 是一个无桥的 $(2t+1)$-正则图且 $ϕ_c(G) \in \{2+\frac{1}{t}, 2 + \frac{2}{2t-1}\}$ 当且仅当 $G$ 存在一个完美匹配 $M$ 使得 $G-M$ 是二部图。这意味着 $G$ 是一个第 1 类图。对于 $t=1$，所有循环流数大于 4 的图都是第 2 类图。我们证明对于所有 $t \geq 1$，有 $2 + \frac{2}{2t-1} = \inf \{ ϕ_c(G)\colon G \text{ 是一个 } (2t+1) \text{-正则第 } 2 \text{ 类图}\}$。这一结论在 [Edge-colorings and circular flow numbers of regular graphs, J. Graph Theory 79 (2015) 1-7] 中被猜想为真。此外，我们证明 $\inf\{ ϕ_c(G)\colon G $ 是一个没有完美匹配（移除该匹配后留下二部图）的 $(2t+1)$-正则第 $1$ 类图$ \} = 2 + \frac{2}{2t-1}$。我们进一步否定了如下猜想：每个 $(2t+1)$-正则第 $1$ 类图的循环流数至多为 $2+\frac{2}{t}$。
英文摘要：
Let $ϕ_c(G)$ be the circular flow number of a bridgeless graph $G$. In [Edge-colorings and circular flow numbers of regular graphs, J. Graph Theory 79 (2015) 1-7] it was proved that, for every $t \geq 1$, $G$ is a bridgeless $(2t+1)$-regular graph with $ϕ_c(G) \in \{2+\frac{1}{t}, 2 + \frac{2}{2t-1}\}$ if and only if $G$ has a perfect matching $M$ such that $G-M$ is bipartite. This implies that $G$ is a class 1 graph. For $t=1$, all graphs with circular flow number bigger than 4 are class 2 graphs. We show for all $t \geq 1$, that $2 + \frac{2}{2t-1} = \inf \{ ϕ_c(G)\colon G \text{ is a } (2t+1) \text{-regular class } 2 \text{ graph}\}$. This was conjectured to be true in [Edge-colorings and circular flow numbers of regular graphs, J. Graph Theory 79 (2015) 1-7]. Moreover we prove that $\inf\{ ϕ_c(G)\colon G $ is a $ (2t+1)$-regular class $1$ graph with no perfect matching whose removal leaves a bipartite graph$ \} = 2 + \frac{2}{2t-1}$. We further disprove the conjecture that every $(2t+1)$-regular class $1$ graph has circular flow number at most $2+\frac{2}{t}$.
Let $k, d$ ($2d \leq k)$ be two positive integers. We generalize the well studied notions of $(k,d)$-colorings and of the circular chromatic number $χ_c$ to signed graphs. This implies a new notion of colorings of signed graphs, and the corresponding chromatic number $χ$. Some basic facts on circular colorings of signed graphs and on the circular chromatic number are proved, and differences to the results on unsigned graphs are analyzed. In particular, we show that the difference between the circular chromatic number and the chromatic number of a signed graph is at most 1. Indeed, there are si…
In this we consider weighted symmetric digraph. Our result generalizes the work of Zhu (J.Comb.Theory, Ser.B, 86 (2002) 109-113) concerning the (k,d)-coloring of a graph, and thus is also a generalization of a corresponding result of Tuza (J.Comb.Theory, Ser.B, 55 (1992) 236-243). Our result also strengthens a result of Goddyn, Tarsi and Zhang (J.Graph Theory 28 (1998) 155-161) concerning the relation between orientation and the (k,d)-coloring of a graph.
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 $p\ge 5$, $$p^2-\frac{5}{2}p+\frac{3}{2}\le f(p)\le 2p^2+2p-5.$$ …
We consider circular version of the famous Nelson-Hadwiger problem. It is know that 4 colors are necessary and 7 colors suffice to color the euclidean plane in such a way that points at distance one get different colors. In $r$-circular coloring we assign arcs of length one of a circle with a perimeter $r$ in such a way that points at distance one get disjoint arcs. In this paper we show the existence of $r$-circular coloring for $r=4+\frac{4\sqrt{3}}{3}\approx 6.30$. It is the first result with $r$-circular coloring of the plane with $r$ smaller than 7. We also show $r$-circular coloring of t…
In this paper, we investigate circular chromatic number of Mycielski construction of graphs. It was shown in \cite{MR2279672} that $t^{\rm th}$ Mycielskian of the Kneser graph $KG(m,n)$ has the same circular chromatic number and chromatic number provided that $m+t$ is an even integer. We prove that if $m$ is large enough, then $χ(M^t(KG(m,n)))=χ_c(M^t(KG(m,n)))$ where $M^t$ is $t^{\rm th}$ Mycielskian. Also, we consider the generalized Kneser graph $KG(m,n,s)$ and show that there exists a threshold $m(n,s,t)$ such that $χ(M^t(KG(m,n,s)))=χ_c(M^t(KG(m,n,s)))$ for $m\geq m(n,s,t)$.
Let $k$ be an positive integer. For any two integers $i$ and $j$ in $\{0,1,\dots,k-1\}$, let $|i-j|_k=\min\{|i-j|,k-|i-j|\}$ be the circular distance between $i$ and $j$. Let $t$ be a nonnegative integer. Suppose $f$ is a mapping from $V(G)$ to $\{0,1,\dots,k-1\}$. If adjacent vertices receive different integers, and for each vertex $u$ of $G$, the number of neighbors $v$ of $u$ with $|f(u)-f(v)|_k=1$ is at most $t$, then $f$ is called a $t$-relaxed 2-distant circular $k$-coloring, or simply a $(\frac{k}{2},t)^*$-coloring of $G$. If $G$ has a $(\frac{k}{2},t)^*$-coloring, then $G$ is called $(…
The notion of the circular coloring of signed graphs is a recent one that simultaneously extends both notions of the circular coloring of graphs and $0$-free coloring of signed graphs. A circular $r$-coloring of a signed graph $(G, σ)$ is to assign points of a circle of circumference $r$, $r\geq 2$, to the vertices of $G$ such that vertices connected by a positive edge are at circular distance at least $1$ and vertices connected by a negative edge are at circular distance at most $\frac{r}{2}-1$. The infimum of all $r$ for which $(G, σ)$ admits a circular $r$-coloring is said to be the circula…
A circular $r$-coloring of a signed graph $(G, σ)$ is an assignment $ϕ$ of points of a circle $C_r$ of circumference $r$ to the vertices of $(G, σ)$ such that for each positive edge $uv$ of $(G, σ)$ the distance of $ϕ(v)$ and $ϕ(v)$ is at least 1 and for each negative edge $uv$ the distance of $ϕ(u)$ from the antipodal of $ϕ(v)$ is at least 1. The circular chromatic number of $(G, σ)$, denoted $χ_c(G, σ)$, is the infimum of $r$ such that $(G, σ)$ admits a circular $r$-coloring. This notion is recently defined by Naserasr, Wang, and Zhu who, among other results, proved that for any signed $d$-d…
A signed graph is a pair $(G, σ)$, where $G$ is a graph and $σ: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, σ)$ a circular $r$-coloring of $(G, σ)$ is an assignment $ψ$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $σ(e)=+$, then $ψ(u)$ and $ψ(v)$ have distance at least $1$, and if $σ(e)=-$, then $ψ(v)$ and the antipodal of $ψ(u)$ have distan…
The main contributions of this paper are three-fold. First, we use a dynamic approach based on Reiter's pioneering work on Karp-Miller computation graphs to give a new and short proof of Mohar's Minty-type Theorem. Second, we bridge circular colorings and discrete event dynamic systems to show that the Barbosa and Gafni's results on circular chromatic number can be generalized to edge-weighted symmetric directed graphs. Third, we use the above-mentioned dynamic view of circular colorings to construct new improved lower bounds on the circular chromatic number of a graph. We show as an example t…
The fractional and circular chromatic numbers are the two most studied non-integral refinements of the chromatic number of a graph. Starting from the definition of a coloring base of a graph, which originated in work related to ergodic theory, we formalize the notion of a gyrocoloring of a graph: the vertices are colored by translates of a single Borel set in the circle group, and neighbouring vertices receive disjoint translates. The corresponding gyrochromatic number of a graph always lies between the fractional chromatic number and the circular chromatic number. We investigate basic propert…
The paper characterizes $(2t+1)$-regular graphs with circular flow number $2 + \frac{2}{2t-1}$. For $t=1$ this is Tutte's characterization of cubic graphs with flow number 4. The class of cubic graphs is the only class of odd regular graphs where a flow number separates the class 1 graphs from the class 2 graphs. We finally state some conjectures and relate them to existing flow-conjectures.
In Raman (2007), using a column construction technique it is proved that every interval graph can be colored online with First Fit with at most $8w(G)$ colors, where $w(G)$ is the size of the maximum clique of $G$. Since the column construction can not be adapted to circular arc graphs we give a different proof to establish an upper bound of $9w(G)$ for online coloring a circular arc graph $G$ with the First Fit algorithm.
Higher-dimensional generalization of Youngs' theorem and circular colorings
College of Liberal Arts and Science, Kitasato University（北里大学文理学院）；Department of Mathematical Sciences, Faculty of Science, Shinshu University（信州大学理学部数学科学系）
We study the infinite graph of $n$-dimensional rectangular grid that doesn't appear distance regular and the distance regular colorings of this graph, which are defined as the distance colorings with respect to completely regular codes. It is proved that the elements of the parameter matrix of an arbitrary distance regular coloring form two monotonic sequences. It is shown that every irreducible distance regular coloring of the $n$-dimensional rectangular grid has at most $2n+1$ colors.
Université Paris Cité（巴黎西岱大学）；CNRS（法国国家科学研究中心）；IRIF（信息与随机研究所）；Zhejiang Normal University（浙江师范大学）；Hainan Bielefeld University of Applied Sciences（海南比勒费尔德应用科技大学）
An $(r-1,1)$-coloring of an $r$-regular graph $G$ is an edge coloring such that each vertex is incident to $r-1$ edges of one color and $1$ edge of a different color. In this paper, we completely characterize all $4$-regular pseudographs (graphs that may contain parallel edges and loops) which do not have a $(3,1)$-coloring. An $\{r-1,1\}$-factor of an $r$-regular graph is a spanning subgraph in which each vertex has degree either $r-1$ or $1$. We prove various conditions that that must hold for any vertex-minimal $5$-regular pseudographs without $(4,1)$-colorings or without $\{4,1\}$-factors.…
Cayley graphs of abelian groups, circular colorings and discrete fundamental groups
College of Liberal Arts and Sciences, Kitasato University（北里大学文理学院）；Department of Mathematical Sciences, Faculty of Science, Shinshu University（信州大学理学部数学科学系）
At most how many (proper) q-colorings does a regular graph admit? Galvin and Tetali conjectured that among all n-vertex, d-regular graphs with 2d|n, none admits more q-colorings than the disjoint union of n/2d copies of the complete bipartite graph K_{d,d}. In this note we give asymptotic evidence for this conjecture, giving an upper bound on the number of proper q-colorings admitted by an n-vertex, d-regular graph of the form a^n b^{n(1+o(1))/d} (where a and b depend on q and where o(1) goes to 0 as d goes to infinity) that agrees up to the o(1) term with the count of q-colorings of n/2d copi…
Given a graph $G$, and a spanning subgraph $H$ of $G$, a circular $q$-backbone $k$-coloring of $(G,H)$ is a proper $k$-coloring $c$ of $G$ such that $q\le \lvert c(u)-c(v)\rvert \le k-q$, for every edge $uv\in E(H)$. The circular $q$-backbone chromatic number of $(G,H)$, denoted by $CBC_q(G,H)$, is the minimum integer $k$ for which there exists a circular $q$-backbone $k$-coloring of $(G,H)$. The Four Color Theorem implies that whenever $G$ is planar, we have $CBC_2(G,H)\le 8$. It is conjectured that this upper bound can be improved to 7 when $H$ is a tree, and to 6 when $H$ is a matching. In …
Using a relation between domination number and incidence chromatic number, we obtain necessary and sufficient conditions for $r$-regular graphs to be $(r+1)$-incidence colorable. Also, we determine the optimal Nordhaus-Gaddum inequality for the incidence chromatic number.
If $G$ is a graph and $\mathcal{H}$ is a set of subgraphs of $G$, we say that an edge-coloring of $G$ is $\mathcal{H}$-polychromatic if every graph from $\mathcal{H}$ gets all colors present in $G$ on its edges. The $\mathcal{H}$-polychromatic number of $G$, denoted $\operatorname{poly}_\mathcal{H} (G)$, is the largest number of colors in an $\mathcal{H}$-polychromatic coloring. In this paper we determine $\operatorname{poly}_\mathcal{H} (G)$ exactly when $G$ is a complete graph on $n$ vertices, $q$ is a fixed nonnegative integer, and $\mathcal{H}$ is one of three families: the family of all m…
A Kempe swap in a proper coloring interchanges the colors on some maximal connected 2-colored subgraph. Two $k$-colorings are $k$-equivalent if we can transform one into the other using Kempe swaps. The triangulated toroidal grid, $T[m\times n]$, is formed from (a toroidal embedding of) the Cartesian product of $C_m$ and $C_n$ by adding parallel diagonals inside all 4-faces. Mohar and Salas showed that not all 4-colorings of $T[m\times n]$ are 4-equivalent. In contrast, Bonamy, Bousquet, Feghali, and Johnson showed that all 6-colorings of $T[m\times n]$ are 6-equivalent. They asked whether the…
We introduce a new graph invariant that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that circular cliques play for the circular chromatic number. The fact that the defined parameter attains on these graphs the `correct' value suggests that the definition is a n…
In this paper, we continue the study of $2$-colorings in hypergraphs. A hypergraph is $2$-colorable if there is a $2$-coloring of the vertices with no monochromatic hyperedge. It is known (see Thomassen [J. Amer. Math. Soc. 5 (1992), 217--229]) that every $4$-uniform $4$-regular hypergraph is $2$-colorable. Our main result in this paper is a strengthening of this result. For this purpose, we define a vertex in a hypergraph $H$ to be a free vertex in $H$ if we can $2$-color $V(H) \setminus \{v\}$ such that every hyperedge in $H$ contains vertices of both colors (where $v$ has no color). We prov…
This paper studies some coloring properties of graph powers. We show that $χ_c(G^{^{\frac{2r+1}{2s+1}}})=\frac{(2s+1)χ_c(G)}{(s-r)χ_c(G)+2r+1}$ provided that $χ_c(G^{^{\frac{2r+1}{2s+1}}})< 4$. As a consequence, one can see that if ${2r+1 \over 2s+1} \leq {χ_c(G) \over 3(χ_c(G)-2)}$, then $χ_c(G^{^{\frac{2r+1}{2s+1}}})=\frac{(2s+1)χ_c(G)}{(s-r)χ_c(G)+2r+1}$. In particular, $χ_c(K_{3n+1}^{^{1\over3}})={9n+3\over 3n+2}$ and $K_{3n+1}^{^{1\over3}}$ has no subgraph with circular chromatic number equal to ${6n+1\over 2n+1}$. This provides a negative answer to a question asked in [Xuding Zhu, Circul…
A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least 2, the neighbors of $v$ receive at least 2 colors. In this paper we present some upper bounds for the dynamic chromatic number of graphs. In this regard, we shall show that there is a constant $c$ such that for every $k$-regular graph $G$, $χ_d(G)\leq χ(G)+ c\ln k +1$. Also, we introduce an upper bound for the dynamic list chromatic number of regular graphs.
Let $R$ be a commutative ring with non-zero identity. The cozero-divisor graph of $R$, denoted by $Γ^{\prime}(R)$, is a graph with vertices in $W^*(R)$, which is the set of all non-zero and non-unit elements of $R$, and two distinct vertices $a$ and $b$ in $W^*(R)$ are adjacent if and only if $a\not\in Rb$ and $b\not\in Ra$. In this paper, we show that the cozero-divisor graph of a von Neumann regular ring with finite clique number is not only weakly perfect but also perfect. Also, an explicit formula for the clique number is given.
For graphs $G$ and $H$, an $H$-coloring of $G$ is a map from the vertices of $G$ to the vertices of $H$ that preserves edge adjacency. We consider the following extremal enumerative question: for a given $H$, which connected $n$-vertex graph with minimum degree $δ$ maximizes the number of $H$-colorings? We show that for non-regular $H$ and sufficiently large $n$, the complete bipartite graph $K_{δ,n-δ}$ is the unique maximizer. As a corollary, for non-regular $H$ and sufficiently large $n$ the graph $K_{k,n-k}$ is the unique $k$-connected graph that maximizes the number of $H$-colorings among …
In 1973, Altshuler characterized the $6$-regular triangulations on the torus to be precisely those that are obtained from a regular triangulation of the $r \times s$ toroidal grid where the vertices in the first and last column are connected by a shift of $t$ vertices. Such a graph is denoted $T(r, s, t)$. In 1999, Collins and Hutchinson classified the $4$-colorable graphs $T(r, s, t)$ with $r, s \geq 3$. In this paper, we point out a gap in their classification and show how it can be fixed. Combined with the classification of the $4$-colorable graphs $T(1, s, t)$ by Yeh and Zhu in 2003, this …
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), …
Let $G$ be a finite $d$-regular graph with a proper edge coloring. An edge Kempe switch is a new proper edge coloring of $G$ obtained by switching the two colors along some bi-chromatic cycle. We prove that any other edge coloring can be obtained by performing finitely many edge Kempe switches, provided that $G$ is replaced with a suitable finite covering graph. The required covering degree is bounded above by a constant depending only on $d$.
Let $H$ be a graph with $Δ(H) \leq 2$, and let $G$ be obtained from $H$ by gluing in vertex-disjoint copies of $K_4$. We prove that if $H$ contains at most one odd cycle of length exceeding $3$, or if $H$ contains at most $3$ triangles, then $χ(G) \leq 4$. This proves the Strong Coloring Conjecture for such graphs $H$. For graphs $H$ with $Δ=2$ that are not covered by our theorem, we prove an approximation result towards the conjecture.
The local chromatic number of a graph was introduced by Erdos et al. in 1986. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan which generalizes the Borsuk-Ulam theo…
The regular graph of ideals of the commutative ring $R$, denoted by $Γ_{reg}(R)$, is a graph whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$ contains a $J$-regular element or $J$ contains an $I$-regular element. In this paper, it is shown that for every Artinian ring $R$, the edge chromatic number of $Γ_{reg}(R)$ equals its maximum degree. Then a formula for the clique number of $Γ_{reg}(R)$ is given. Also, it is proved that for every reduced ring $R$ with $n(\geq3)$ minimal prime ideals, the edge chromat…
Let $G$ be a graph with no isolated vertices. A {\em $k$-coupon coloring} of $G$ is an assignment of colors from $[k] := \{1,2,\dots,k\}$ to the vertices of $G$ such that the neighborhood of every vertex of $G$ contains vertices of all colors from $[k]$. The maximum $k$ for which a $k$-coupon coloring exists is called the {\em coupon coloring number} of $G$, and is denoted $χ_{c}(G)$. In this paper, we prove that every $d$-regular graph $G$ has $χ_{c}(G) \geq (1 - o(1))d/\log d$ as $d \rightarrow \infty$, and the proportion of $d$-regular graphs $G$ for which $χ_c(G) \leq (1 + o(1))d/\log d$ t…
In this article, we study subword complexity of colorings of regular trees. We characterize colorings of bounded subword complexity and study Sturmian colorings, which are colorings of minimal unbounded subword complexity. We classify Sturmian colorings using their type sets. We show that any Sturmian coloring is a lifting of a coloring on a quotient graph of the tree which is a geodesic or a ray with loops possibly attached, thus a lifting of an "infinte word". We further give a complete characterization of the quotient graph for eventually periodic ones.
DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvořák and Postle. Many known upper bounds for the list-chromatic number extend to the DP-chromatic number, but not all of them do. In this note we describe some unusual properties of DP-coloring that set it aside from list coloring. In particular, we give an example of a planar bipartite graph with DP-chromatic number $4$ and prove that the edge-DP-chromatic number of a $d$-regular graph with $d\geq 2$ is always at least $d+1$.
Given a perfect coloring of a graph, we prove that the $L_1$ distance between two rows of the adjacency matrix of the graph is not less than the $L_1$ distance between the corresponding rows of the parameter matrix of the coloring. With the help of an algebraic approach, we deduce corollaries of this result for perfect $2$-colorings, perfect colorings in distance-$l$ graphs and in distance-regular graphs. We also provide examples when the obtained property reject several putative parameter matrices of perfect colorings in infinite graphs.
For graphs $G$ and $H$, an $H$-coloring of $G$ is an adjacency preserving map from the vertices of $G$ to the vertices of $H$. $H$-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of $H$-colorings. In this paper, we prove several results in this area. First, we find a class of graphs ${\mathcal H}$ with the property that for each $H \in {\mathcal H}$, the $n$-vertex tree that minimizes the number of $H$-colorings is …
For an $r$-regular graph $G$, we define an edge-coloring $c$ with colors from $\{1,2,\cdots,$ $k\}$, in such a way that any vertex of $G$ is incident to at least one edge of each color. The multiset-color $c_m(v)$ of a vertex $v$ is defined as the ordered tuple $(a_1,a_2,\cdots ,a_k)$, where $a_i \ (1\leq i\leq k)$ denotes the number of edges with color $i$ which are incident with $v$ in $G$. Then this edge-coloring $c$ is called a {\it $k$-kaleidoscopic coloring} of $G$ if every two distinct vertices in $G$ have different multiset-colors and in this way the graph $G$ is defined as a {\it $k$-…
Given an edge-coloring of a graph $G$, we associate to every vertex $v$ of $G$ the set of colors appearing on the edges incident with $v$. The palette index of $G$ is defined as the minimum number of such distinct sets, taken over all possible edge-colorings of $G$. A graph with a small palette index admits an edge-coloring which can be locally considered to be almost symmetric, since few different sets of colors appear around its vertices. Graphs with palette index $1$ are $r$-regular graphs admitting an $r$-edge-coloring, while regular graphs with palette index $2$ do not exist. Here, we cha…
