Homomorphism complexes and k-cores

For any fixed graph $G$, we prove that the topological connectivity of the graph homomorphism complex Hom($G,K_m$) is at least $m?D\left(G\right)?2$, where $D\left(G\right)={max}_{H?G}\delta \left(H\right)$, for $\delta \left(H\right)$ the minimum degree of a vertex in a subgraph $H$. This generalizes a theorem of C?uki? and Kozlov, in which the maximum degree $\Delta \left(G\right)$ was used in place of $D\left(G\right)$, and provides a high-dimensional analogue of the graph theoretic bound for chromatic number, $\chi \left(G\right)\le D\left(G\right)+1$, as $\chi \left(G\right)=min\{m:\text{Hom}(G,K_m)\ne ?\}$. Furthermore, we use this result to examine homological phase transitions in the random polyhedral complexes Hom$(G(n,p),K_m)$ when $p=c∕n$ for a fixed constant $c>0$.     

The list L(2,1)-labeling of planar graphs

Let $\mathbf{N}$ be the set of all positive integers. A list assignment of a graph $G$ is a function $L:V\left(G\right)?2^{\mathbf{N}}$ that assigns each vertex $v$ a list $L\left(v\right)$ for all $v\in V\left(G\right)$. We say that $G$ is $L$-$(2,1)$-choosable if there exists a function $?$ such that $?\left(v\right)\in L\left(v\right)$ for all $v\in V\left(G\right)$, $|?\left(u\right)??\left(v\right)|\ge 2$ if $u$ and $v$ are adjacent, and $|?\left(u\right)??\left(v\right)|\ge 1$ if $u$ and $v$ are at distance 2. The list-$L(2,1)$-labeling number ${\lambda }_l\left(G\right)$ of $G$ is the minimum $k$ such that for every list assignment $L=\{L\left(v\right):|L\left(v\right)|=k,v\in V\left(G\right)\}$, $G$ is $L$-$(2,1)$-choosable. We prove that if $G$ is a planar graph with girth $g\ge 8$ and its maximum degree $\Delta$ is large enough, then ${\lambda }_l\left(G\right)\le \Delta +3$. There are graphs with large enough $\Delta$ and $g\ge 8$ having ${\lambda }_l\left(G\right)=\Delta +3$.     

Classification of some 3-subgroups of the finite groups of Lie type E6

We consider the finite exceptional group of Lie type $G=E_6^{\epsilon }(q)$ (universal version) with $3|q?\epsilon$, where $E_6^{+1}(q)=E_6(q)$ and $E_6^{?1}(q){=}^2{E}_6(q)$. We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local $M<G$ which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing $Z(G)$, all extraspecial subgroups containing $Z(G)$, and all cyclic groups of order 9 containing $Z(G)$. These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.     

A new approach to the chromatic number of the square of Kneser graph K(2k+1,k)

The vertices of Kneser graph $K(n,k)$ are the subsets of $\{1,2,\dots ,n\}$ of cardinality $k$, two vertices are adjacent if and only if they are disjoint. The square $G^2$ of a graph $G$ is defined on the vertex set of $G$ with two vertices adjacent if their distance in $G$ is at most 2. Z. Füredi, in 2002, proposed the problem of determining the chromatic number of the square of the Kneser graph. The first non-trivial problem arises when $n=2k+1$. It is believed that $\chi \left(K^2(2k+1,k)\right)=2k+c$ where $c$ is a constant, and yet the problem remains open. The best known upper bounds are by Kim and Park: $8k∕3+20∕3$ for 1$k\ge 3$ (Kim and Park, 2014) and $32k∕15+32$ for $k\ge 7$ (Kim and Park, 2016). In this paper, we develop a new approach to this coloring problem by employing graph homomorphisms, cartesian products of graphs, and linear congruences integrated with combinatorial arguments. These lead to $\chi \left(K^2(2k+1,k)\right)\le 5k∕2+c$, where $c$ is a constant in $\{5∕2,9∕2,5,6\}$, depending on $k\ge 2$.     

On the spanning connectivity of the generalized Petersen graphs P(n,3)

In this paper, we employed lattice model to describe the three internally vertex-disjoint paths that span the vertex set of the generalized Petersen graph $P(n,3)$. We showed that the $P(n,3)$ is 3-spanning connected for odd $n$. Based on the lattice model, five amalgamated and one extension mechanisms are introduced to recursively establish the 3-spanning connectivity of the $P(n,3)$. In each amalgamated mechanism, a particular lattice trail was amalgamated with the lattice trails that was dismembered, transferred, or extended from parts of the lattice trails for $P(n?6,3)$, where a lattice tail is a trail in the lattice model that represents a path in $P(n,3)$.     

d-matching in 3-uniform hypergraphs

A matching in a 3-uniform hypergraph is a set of pairwise disjoint edges. A $d$-matching in a 3-uniform hypergraph $H$ is a matching of size $d$. Let $V_1,V_2$ be a partition of $n$ vertices such that $\left|V_1\right|=2d?1$ and $\left|V_2\right|=n?2d+1$. Denote by $E_3(2d?1,n?2d+1)$ the 3-uniform hypergraph with vertex set $V_1\cup V_2$ consisting of all those edges which contain at least two vertices of $V_1$. Let $H$ be a 3-uniform hypergraph of order $n\ge 9d^2$ such that $deg\left(u\right)+deg\left(v\right)>2\left[\left(\genfrac{}{}{0ex}{}{n?1}{2}\right)?\left(\genfrac{}{}{0ex}{}{n?d}{2}\right)\right]$ for any two adjacent vertices $u,v\in V\left(H\right)$. In this paper, we prove $H$ contains a $d$-matching if and only if $H$ is not a subgraph of $E_3(2d?1,n?2d+1)$.     

Properties of minimally t-tough graphs

A graph $G$ is minimally $t$-tough if the toughness of $G$ is $t$ and the deletion of any edge from $G$ decreases the toughness. Kriesell conjectured that for every minimally $1$-tough graph the minimum degree $\delta \left(G\right)=2$. We show that in every minimally $1$-tough graph $\delta \left(G\right)\le \frac{n}{3}+1$. We also prove that every minimally $1$-tough, claw-free graph is a cycle. On the other hand, we show that for every positive rational number $t$ any graph can be embedded as an induced subgraph into a minimally $t$-tough graph.     

An improved upper bound on the adjacent vertex distinguishing total chromatic number of graphs

An adjacent vertex distinguishing total $k$-coloring of a graph $G$ is a proper total $k$-coloring of $G$ such that any pair of adjacent vertices have different sets of colors. The minimum number $k$ needed for such a total coloring of $G$ is denoted by ${\chi }_a^{\prime \prime }\left(G\right)$. In this paper we prove that ${\chi }_a^{\prime \prime }\left(G\right)\le 2\Delta \left(G\right)?1$ if $\Delta \left(G\right)\ge 4$, and ${\chi }_a^{\prime \prime }\left(G\right)\le ?\frac{5\Delta \left(G\right)+8}{3}?$ in general. This improves a result in Huang et al. (2012) which states that ${\chi }_a^{\prime \prime }\left(G\right)\le 2\Delta \left(G\right)$ for any graph with $\Delta \left(G\right)\ge 3$.     

A characterization for the neighbor-distinguishing total chromatic number of planar graphs with Δ=13

The neighbor-distinguishing total chromatic number ${\chi }_a^{\prime \prime }\left(G\right)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be totally colored using $k$ colors with a condition that any two adjacent vertices have different sets of colors. In this paper, we give a sufficient and necessary condition for a planar graph $G$ with maximum degree 13 to have ${\chi }_a^{\prime \prime }\left(G\right)=14$ or ${\chi }_a^{\prime \prime }\left(G\right)=15$. Precisely, we show that if $G$ is a planar graph of maximum degree 13, then $14\le {\chi }_a^{\prime \prime }\left(G\right)\le 15$; and ${\chi }_a^{\prime \prime }\left(G\right)=15$ if and only if $G$ contains two adjacent 13-vertices.     

List 3-dynamic coloring of graphs with small maximum average degree

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that for any vertex $v$, there are at least $min\{r,{deg}_G\left(v\right)\}$ distinct colors in $N_G\left(v\right)$. The $r$-dynamic chromatic number${\chi }_r^d\left(G\right)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$. The list$r$-dynamic chromatic number of a graph $G$ is denoted by $c{h}_r^d\left(G\right)$.Recently, Loeb et al. (0000) showed that the list $3$-dynamic chromatic number of a planar graph is at most 10. And Cheng et al. (0000) studied the maximum average condition to have ${\chi }_3^d\left(G\right)\le 4,5$, or $6$. On the other hand, Song et al. (2016) showed that if $G$ is planar with girth at least 6, then ${\chi }_r^d\left(G\right)\le r+5$ for any $r\ge 3$.In this paper, we study list 3-dynamic coloring in terms of maximum average degree. We show that $c{h}_3^d\left(G\right)\le 6$ if $mad\left(G\right)<\frac{18}{7}$, $c{h}_3^d\left(G\right)\le 7$ if $mad\left(G\right)<\frac{14}{5}$, and $c{h}_3^d\left(G\right)\le 8$ if $mad\left(G\right)<3$. All of the bounds are tight.     

A degree condition for a graph to have (a,b)-parity factors

Let $a$ and $b$ be two positive integers such that $a\le b$ and $a\equiv b(mod2)$. A graph $F$ is an $(a,b)$-parity factor of a graph $G$ if $F$ is a spanning subgraph of $G$ and for all vertices $v\in V\left(F\right)$, $d_F\left(v\right)\equiv b(mod2)$ and $a\le d_F\left(v\right)\le b$. In this paper we prove that every connected graph $G$ with $n\ge b(a+b)(a+b+2)∕\left(2a\right)$ vertices has an $(a,b)$-parity factor if $na$ is even, $\delta \left(G\right)\ge (b?a)∕a+a$, and for any two nonadjacent vertices $u,v\in V\left(G\right)$, $max\{d_G\left(u\right),d_G\left(v\right)\}\ge \frac{an}{a+b}$. This extends an earlier result of Nishimura (1992) and strengthens a result of Cai and Li (1998).     

List star chromatic index of sparse graphs

A star edge-coloring of a graph $G$ is a proper edge coloring such that every 2-colored connected subgraph of $G$ is a path of length at most 3. For a graph $G$, let the list star chromatic index of $G$, $c{h}_s^{\prime }\left(G\right)$, be the minimum $k$ such that for any $k$-uniform list assignment $L$ for the set of edges, $G$ has a star edge-coloring from $L$. Dvo?ák et al. (2013) asked whether the list star chromatic index of every subcubic graph is at most 7. In Kerdjoudj et al. (2017) we proved that it is at most 8. In this paper we consider graphs with any maximum degree, we proved that if the maximum average degree of a graph $G$ is less than $\frac{14}{5}$ (resp. 3), then $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+2$ (resp. $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+3$).