Proper connection and size of graphs

An edge-coloured graph $G$ is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a connected graph $G,$ denoted by $\mathrm{pc}\left(G\right)$, is the smallest number of colours that are needed in order to make $G$ properly connected. Our main result is the following: Let $G$ be a connected graph of order $n$ and $k\ge 2$. If $\left|E\left(G\right)\right|\ge \left(\genfrac{}{}{0ex}{}{n?k?1}{2}\right)+k+2$, then $\mathrm{pc}\left(G\right)\le k$ except when $k=2$ and $G\in \{G_1,G_2\},$ where $G_1=K_1\vee (2K_1+K_2)$ and $G_2=K_1\vee (K_1+2K_2).$     

Nice derivations over principal ideal domains

In this paper we investigate to what extent the results of Z. Wang and D. Daigle on “nice derivations” of the polynomial ring $k[X,Y,Z]$ over a field k of characteristic zero extend to the polynomial ring $R[X,Y,Z]$ over a PID R, containing the field of rational numbers. One of our results shows that the kernel of a nice derivation on $k[X_1,X_2,X_3,X_4]$ of rank at most three is a polynomial ring over k.     

Sum choice number of generalized θ-graphs

Let $G=\left(VE\right)$ be a simple graph and for every vertex $v\in V$ let $L\left(v\right)$ be a set (list) of available colors. $G$ is called $L$-colorable if there is a proper coloring $\phi$ of the vertices with $\phi \left(v\right)\in L\left(v\right)$ for all $v\in V$. A function $f:V\to \mathbb{N}$ is called a choice function of $G$ and $G$ is said to be $f$-list colorable if $G$ is $L$-colorable for every list assignment $L$ choice function is defined by $size\left(f\right)={\sum }_{v\in V}f\left(v\right)$ and the sum choice number ${\chi }_{sc}\left(G\right)$ denotes the minimum size of a choice function of $G$.Sum list colorings were introduced by Isaak in 2002 and got a lot of attention since then.For $r\ge 3$ a generalized ${\theta }_{k_1{k}_2\dots k_r}$-graph is a simple graph consisting of two vertices $v_1$ and $v_2$ connected by $r$ internally vertex disjoint paths of lengths $k_1,k_2,\dots ,k_r$ $(k_1\le k_2\le ?\le k_r)$.In 2014, Carraher et al. determined the sum-paintability of all generalized $\theta$-graphs which is an online-version of the sum choice number and consequently an upper bound for it.In this paper we obtain sharp upper bounds for the sum choice number of all generalized $\theta$-graphs with $k_1\ge 2$ and characterize all generalized $\theta$-graphs $G$ which attain the trivial upper bound $\left|V\left(G\right)\right|+\left|E\left(G\right)\right|$.     

On vertex-disjoint cycles and degree sum conditions

This paper considers a degree sum condition sufficient to imply the existence of $k$ vertex-disjoint cycles in a graph $G$. For an integer $t\ge 1$, let ${\sigma }_t\left(G\right)$ be the smallest sum of degrees of $t$ independent vertices of $G$. We prove that if $G$ has order at least $7k+1$ and ${\sigma }_4\left(G\right)\ge 8k?3$, with $k\ge 2$, then $G$ contains $k$ vertex-disjoint cycles. We also show that the degree sum condition on ${\sigma }_4\left(G\right)$ is sharp and conjecture a degree sum condition on ${\sigma }_t\left(G\right)$ sufficient to imply $G$ contains $k$ vertex-disjoint cycles for $k\ge 2$.     

The confirmation of a conjecture on disjoint cycles in a graph

Let $t$ and $k$ be two integers with $t\ge 5$ and $k\ge 2$. For a graph $G$ and a vertex $x$ of $G$, we use $d_G\left(x\right)$ to denote the degree of $x$ in $G$. Define ${\sigma }_t\left(G\right)$ to be the minimum value of ${\sum }_{x\in X}{d}_G\left(x\right)$, where $X$ is an independent set of $G$ with $\left|X\right|=t$. This paper proves the following conjecture proposed by Gould et al. (2018). If $G$ is a graph of sufficiently large order with ${\sigma }_t\left(G\right)\ge 2kt?t+1$, then $G$ contains $k$ vertex-disjoint cycles.     

Turán numbers of vertex-disjoint cliques in r-partite graphs

For two graphs $G$ and $H$, the Turán number$ex(G,H)$ is the maximum number of edges in a subgraph of $G$ that contains no copy of $H$. Chen, Li, and Tu determined the Turán numbers $ex(K_{m,n},k{K}_2)$ for all $k\ge 1$ Chen et al. (2009). In this paper we will determine the Turán numbers $ex(K_{a_1,\dots ,a_r},k{K}_r)$ for all $r\ge 3$ and $k\ge 1$.     

Partitions of natural numbers with the same weighted representation functions

TextFor any given two positive integers $k_1$ and $k_2$, and any set A of nonnegative integers, let $r_{k_1,k_2}(A,n)$ denote the number of solutions of the equation $n=k_1{a}_1+k_2{a}_2$ with $a_1,a_2\in A$. In this paper, we determine all pairs $k_1,k_2$ of positive integers for which there exists a set $A?\mathbb{N}$ such that $r_{k_1,k_2}(A,n)=r_{k_1,k_2}(\mathbb{N}?A,n)$ for all $n?n_0$. We also pose several problems for further research.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EnezEsJl0OY.     

Proof of a conjecture on monomial graphs

Let e be a positive integer, p be an odd prime, $q=p^e$, and ${\mathbb{F}}_q$ be the finite field of q elements. Let $f,g\in {\mathbb{F}}_q[X,Y]$. The graph $G_q(f,g)$ is a bipartite graph with vertex partitions $P={\mathbb{F}}_q^3$ and $L={\mathbb{F}}_q^3$, and edges defined as follows: a vertex $(p)=(p_1,p_2,p_3)\in P$ is adjacent to a vertex $[l]=[l_1,l_2,l_3]\in L$ if and only if $p_2+l_2=f(p_1,l_1)$ and $p_3+l_3=g(p_1,l_1)$. If $f=XY$ and $g=X{Y}^2$, the graph $G_q(XY,X{Y}^2)$ contains no cycles of length less than eight and is edge-transitive. Motivated by certain questions in extremal graph theory and finite geometry, people search for examples of graphs $G_q(f,g)$ containing no cycles of length less than eight and not isomorphic to the graph $G_q(XY,X{Y}^2)$, even without requiring them to be edge-transitive. So far, no such graphs $G_q(f,g)$ have been found. It was conjectured that if both f and g are monomials, then no such graphs exist. In this paper we prove the conjecture.     

On degree sum conditions for 2-factors with a prescribed number of cycles

For a vertex subset $X$ of a graph $G$, let ${\Delta }_t\left(X\right)$ be the maximum value of the degree sums of the subsets of $X$ of size $t$. In this paper, we prove the following result: Let $k,m$ be positive integers, and let $G$ be an $m$-connected graph of order $n\ge 5k?2$. If ${\Delta }_2\left(X\right)\ge n$ for every independent set $X$ of size $?m∕k?+1$ in $G$, then $G$ has a 2-factor with exactly $k$ cycles. This is a common extension of the results obtained by Brandt et al. (1997) and Yamashita (2008), respectively.     

Hamiltonian cycles with all small even chords

Let $G$ be a graph of order $n\ge 3$. An even squared Hamiltonian cycle (ESHC) of $G$ is a Hamiltonian cycle $C=v_1{v}_2\dots v_n{v}_1$ of $G$ with chords $v_i{v}_{i+3}$ for all $1\le i\le n$ (where $v_{n+j}=v_j$ for $j\ge 1$). When $n$ is even, an ESHC contains all bipartite $2$-regular graphs of order $n$. We prove that there is a positive integer $N$ such that for every graph $G$ of even order $n\ge N$, if the minimum degree is $\delta \left(G\right)\ge \frac{n}{2}+92$, then $G$ contains an ESHC. We show that the condition of $n$ being even cannot be dropped and the constant $92$ cannot be replaced by $1$. Our results can be easily extended to even $k$th powered Hamiltonian cycles for all $k\ge 2$.     

Packing chromatic number of cubic graphs

A packing$k$-coloring of a graph $G$ is a partition of $V\left(G\right)$ into sets $V_1,\dots ,V_k$ such that for each $1\le i\le k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, ${\chi }_p\left(G\right)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. Sloper showed that there are $4$-regular graphs with arbitrarily large packing chromatic number. The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We answer this question in the negative. Moreover, we show that for every fixed $k$ and $g\ge 2k+2$, almost every $n$-vertex cubic graph of girth at least $g$ has the packing chromatic number greater than $k$.     

Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

For bipartite graphs $G_1,G_2,\dots ,G_k$, the bipartite Ramsey number $b(G_1,G_2,\dots ,G_k)$ is the least positive integer $b$ so that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a copy of $G_i$ in the $i$th color for some $i$. In this paper, our main focus will be to bound the following numbers: $b(C_{2t_1},C_{2t_2})$ and $b(C_{2t_1},C_{2t_2},C_{2t_3})$ for all $t_i\ge 3,$$b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4})$ for $3\le t_i\le 9,$ and $b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4},C_{2t_5})$ for $3\le t_i\le 5.$ Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result.