Motivated by the plane coloring problem, Eggleton, Erd\H{o}s and Skelton initiated the study of distance graphs. Let $D$ be a set of positive integers. The distance graph generated by $D$, denoted by $G(\mathbb{Z}, D)$, has all integers $\mathbb{Z}$ as the vertex set, and two vertices $x$ and $y$ are adjacent whenever $|x-y| \in D$. The chromatic number, circular chromatic number and fractional chromatic number of distance graphs have been studied extensively in the past two decades; these coloring parameters are also closely related to some problems studied in number theory and geometry. We survey some research advances and open problems on coloring parameters of distance graphs.

For an edge $xy$, let $M(xy)$ be the maximum of the degrees of $x$ and $y$. The {\em minimax degree} (or $M$-degree) of a graph $G$ is $M^*(G)=\min\{M(xy)| xy\in E(G)\}$. In order to get upper bounds on the game chromatic number of planar graphs, He, Hou, Lih, Shao, Wang, and Zhu showed that every planar graph $G$ without leaves and $4$-cycles has minimax degree at most $8$, which was improved by Borodin, Kostochka, Sheikh, and Yu to the sharp bound $7$. We show that every planar graph $G$ without leaves and $4$- and $5$-cycles has $M$-degree at most $5$, which bound is sharp. We also show that every planar graph $G$ without leaves and cycles of length from $4$ to $7$ has $M$-degree at most $4$, which bound is attained even on planar graphs with no cycles of length from $4$ to arbitrarily large number. Besides, we give sufficient conditions for a planar graph to have $M$-degrees $3$ and $2$. Similar results are obtained for graphs embeddable into the projective plane, the torus and the Klein bottle.

An $m$-coloring of a graph $G$ is a mapping $f:V(G)\rightarrow \{1,2,\ldots,m\}$ such that $f(x)\neq f(y)$ for any two adjacent vertices $x$ and $y$ in $G$. The chromatic number $\chi(G)$ of $G$ is the minimum number $m$ such that $G$ is $m$-colorable. An equitable $m$-coloring of a graph $G$ is an $m$-coloring $f$ such that any two color classes differ in size by at most one. The equitable chromatic number $\chi_{_=}(G)$ of $G$ is the minimum number $m$ such that $G$ is equitably $m$-colorable. The equitable chromatic threshold $\chi_{_=}^*(G)$ of $G$ is the minimum number $m$ such that $G$ is equitably $r$-colorable for all $r\geq m$. It is clear that $\chi(G)\leq \chi_{_=}(G) \leq \chi_{_=}^*(G)$. For $n\geq 2k+1$, the Kneser graph $\sf KG(n,k)$ has the vertex set consisting of all $k$-subsets of an $n$-set. Two distinct vertices are adjacent in $\sf KG(n,k)$ if they have empty intersection as subsets. The Kneser graph $\sf KG(2k+1,k)$ is called the Odd graph, denoted by $O_k$. In this paper, we study the equitable colorings of Kneser graphs $\sf KG(n,k)$. Mainly, we obtain that $\chi_{_=}(\sf KG(n,k))\leq \chi_{_=}^*(\sf KG(n,k))\leq n-k+1$ and $\chi(O_k)=\chi_{_=}(O_k)=\chi_{_=}^*(O_k)=3$. We also show that $\chi_{_=}(\sf KG(n,k))=\chi_{_=}^*(\sf KG(n,k))$ for $k=2$ or 3 and obtain their exact values.

There are several works [6] (and [13]), [8], [2] and [14] enumerating four-dimensional parallelotopes. Engel [9] was the first who distinguished 17 zonotopal parallelotopes among them. Each zonotopal parallelotope is the Minkowski sum of segments whose generating vectors form a unimodular system. We show that there are exactly 17 four-dimensional unimodular systems. Hence there are 17 four-dimensional zonotopal parallelotopes. We prove that other 35 four-dimensional parallelotopes are: the regular 24-cell $\{3,4,3\}$ and 34 sums of the 24-cell with non-zero zonotopal parallelotopes. We give a detailed description of the construction of these 35 parallelotopes.

Let $k, n_1, \dots, n_k$ be fixed positive integers, $c_1, \dots, c_k \in GF(q)^*$, and $a_1, \dots, a_k, c \in GF(q)$. We study the number of solutions in $GF(q)$ of the equation $c_1D_{n_1}(x_1, a_1) + c_2D_{n_2}(x_2, a_2) + \cdots + c_kD_{n_k}(x_k, a_k) = c$, where each $D_{n_i}(x_i, a_i)$, $1 \leq i \leq k$, is the Dickson polynomial of degree $n_i$ with parameter $a_i$. We also employ the results of the $k = 1$ case to recover the cardinality of preimages and images of Dickson polynomials obtained earlier by Chou, Gomez-Calderon and Mullen [1].

This paper investigates the problem of factoring $K_{2n}-I_n$ into 2-factors of two kinds or three kinds: (1) $C_t$-factors and $C_{2t}$-factors, (2) $C_4$-factors and $C_{2t}$-factors, (3) $C_4$-factors, $C_8$-factors and $C_{16}$-factors.

In the western world there is this myth that the ancient Chinese knew a special case of Fermat's Little Theorem and erroneously took it as a criterion for primality, namely, that $n$ is a prime if and only if $2^{n-]} -1$ is divisible by $n$. This article discusses how this myth might have come about, in particular tells the story of an investigation on number theory by Li Shanlan in the mid $19^{\rm th}$ century. The discussion touches upon the social history of the incident in connection with the polarized attitude different foreigners took towards Chinese mathematics at the time.

This paper considers list circular colouring of graphs in which the colour list assigned to each vertex is an interval of a circle. The {\em circular consecutive choosability} $ch_{cc}(G)$ of $G$ is defined to be the least $t$ such that for any circle $S(r)$ of length $r \geq \chi_c(G)$, if each vertex $x$ of $G$ is assigned an interval $L(x)$ of $S(r)$ of length $t$, then there is a circular $r$-colouring $f$ of $G$ such that $f(x) \in L(x)$. We show that for any finite graph $G$, $\chi(G)-1 \leq ch_{cc}(G) \lt 2 \chi_c(G)$. We determine the value of $ch_{cc}(G)$ for complete graphs, trees, even cycles and balanced complete bipartite graphs. Upper and lower bounds for $ch_{cc}(G)$ are given for some other classes of graphs.

In the related literatures, the eccentricities of graphs have been studied recently. The main purpose of this paper is to discuss the eccentric spectrum of a graph. For any two vertices $u$ and $v$ in a connected graph $G$, $d_G(u,v)$ denotes the distance between vertices $u$ and $v$. The eccentricity $e_G(v)$ of a vertex $v$ in $G$ is the maximum number of $d_G(v,u)$ over all vertex $u$. A vertex $u$ is an eccentric vertex if there exists a vertex $v$ such that $e_G(v)=d_G(v,u)$. A number $k$ is called an eccentric number of $G$ if, for each vertex $v$ with $e_G(v)=k$, $v$ is an eccentric vertex. The eccentric spectrum $S_G$ of a connected graph $G$ is a set of all eccentric numbers in $G$. If $d$ is the diameter of $G$, then $d\in S_G$. In the paper, we show that for positive integers $r\leq d\leq 2r$ and $d\in S\subseteq \{r,r+1,...,d\}$, there exists a connected graph $G$ with radius $r$, diameter $d$ and eccentric spectrum $S$. This result also proves the conjecture of Chartrand, Gu, Schultz, and Winters in [4].

An adjacent vertex distinguishing total coloring of a graph $G$ is a proper total coloring of $G$ such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing total coloring of $G$ is denoted by $\chi''_{a}(G)$. Let mad$(G)$ and $\Delta(G)$ denote the maximum average degree and the maximum degree of a graph $G$, respectively. In this paper, we prove the following results: (1) If $G$ is a graph with mad$(G)\lt 3$ and $\Delta(G)\ge 5$, then $\Delta(G)+1\le \chi''_{a}(G)\le \Delta(G)+2$, and $\chi''_{a}(G)=\Delta(G)+2$ if and only if $G$ contains two adjacent vertices of maximum degree; (2) If $G$ is a graph with mad$(G)\lt 3$ and $\Delta(G)\le 4$, then $\chi''_{a}(G)\le 6$; (3) If $G$ is a graph with mad$(G)\lt \frac 83$ and $\Delta(G)\le 3$, then $\chi''_{a}(G)\le 5$.

We present a new class of LYM orders, which generalizes Lih’s result and is a common generalization of Griggs’ result and a result of West, Harper and Daykin.

The circular chromatic number $\chi_c(G)$ of a graph $G$ is a very natural generalization of the concept of chromatic number $\chi(G)$, and has been studied extensively in the past decade. In this paper we present a new method for bounding the circular chromatic number from below. Let $\omega$ be an acyclic orientation of a graph $G$. A sequence of acyclic orientations $\omega_1$, $\omega_2$, $\omega_3,\ldots$ is obtained from $\omega$ in such a way that $\omega_1=\omega$, and $\omega_i$ ($i\geq 2$) is obtained from $\omega_{i-1}$ by reversing the orientations of the edges incident to the sinks of $w_{i-1}$. This sequence is completely determined by $\omega$, and it can be proved that there are positive integers $p$ and $M$ such that $\omega_i=\omega_{i+p}$ for every integer $i\geq M$. The value $p$ at its minimum is denoted by $p_\omega$. To bound $\chi_c(G)$ from below, the methodology we develop in this paper is based on the acyclic orientations $\omega_M, \omega_{M+1},\cdots,\omega_{M+p_\omega-1}$ of $G$. Our method demonstrates for the first time the possibility of extracting some information about $\chi_c(G)$ from the period $\omega_M, \omega_{M+1},\cdots,\omega_{M+p_\omega-1}$ to derive lower bounds for $\chi_c(G)$.

We sketch the ideas in the proofs of results on Ramsey numbers of a cycle, particularly in many colors, in which one is due to professor Ko-Wei Lih and the author for the right order of magnitude of Ramsey number $r_k(C_{2m})$ as $k\to\infty$ for $m=2,3,5$.