Defective 2-colorings of planar graphs without 4-cycles and 5-cycles

A 2-coloring is a coloring of vertices of a graph with colors 1 and 2. Define $V_i?\{v\in V\left(G\right):c\left(v\right)=i\}$ for $i=1$ and $2.$ We say that $G$ is $(d_1,d_2)$-colorable if $G$ has a 2-coloring such that $V_i$ is an empty set or the induced subgraph $G\left[V_i\right]$ has the maximum degree at most $d_i$ for $i=1$ and $2.$ Let $G$ be a planar graph without 4-cycles and 5-cycles. We show that the problem to determine whether $G$ is $(0,k)$-colorable is NP-complete for every positive integer $k.$ Moreover, we construct non-$(1,k)$-colorable planar graphs without 4-cycles and 5-cycles for every positive integer $k.$ In contrast, we prove that $G$ is $(d_1,d_2)$-colorable where $(d_1,d_2)=(4,4),(3,5),$ and $(2,9).$     

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.     

Maximum average degree and relaxed coloring

We say a graph is $(d,d,\dots ,d,0,\dots ,0)$-colorable with $a$ of $d$’s and $b$ of $0$’s if $V\left(G\right)$ may be partitioned into $b$ independent sets $O_1,O_2,\dots ,O_b$ and $a$ sets $D_1,D_2,\dots ,D_a$ whose induced graphs have maximum degree at most $d$. The maximum average degree, $mad\left(G\right)$, of a graph $G$ is the maximum average degree over all subgraphs of $G$. In this note, for nonnegative integers $a,b$, we show that if $mad\left(G\right)<\frac{4}{3}a+b$, then $G$ is $(1_1,1_2,\dots ,1_a,0_1,\dots ,0_b)$-colorable.     

On disjoint matchings in cubic graphs

For $i=2,3$ and a cubic graph $G$ let ${\nu }_i\left(G\right)$ denote the maximum number of edges that can be covered by $i$ matchings. We show that ${\nu }_2\left(G\right)\ge \frac{4}{5}\left|V\left(G\right)\right|$ and ${\nu }_3\left(G\right)\ge \frac{7}{6}\left|V\left(G\right)\right|$. Moreover, it turns out that ${\nu }_2\left(G\right)\le \frac{\left|V\left(G\right)\right|+2{\nu }_3\left(G\right)}{4}$.     

Approximately locating an invisible agent in a graph with relative distance queries

In a pursuit evasion game on a finite, simple, undirected, and connected graph $G$, a first player visits vertices $m_1,m_2,\dots$ of $G$, where $m_{i+1}$ is in the closed neighborhood of $m_i$ for every $i$, and a second player probes arbitrary vertices $c_1,c_2,\dots$ of $G$, and learns whether or not the distance between $c_{i+1}$ and $m_{i+1}$ is at most the distance between $c_i$ and $m_i$. Up to what distance $d$ can the second player determine the position of the first? For trees of bounded maximum degree and grids, we show that $d$ is bounded by a constant. We conjecture that $d=O(logn)$ for every graph $G$ of order $n$, and show that $d=0$ if $m_{i+1}$ may differ from $m_i$ only if $i$ is a multiple of some sufficiently large integer.     

A note on conservative galaxies,Skolem systems,cyclic cycle decompositions,and Heffter arrays

The conservative number of a graph $G$ is the minimum positive integer $M$, such that $G$ admits an orientation and a labeling of its edges by distinct integers in $\{1,2,\dots ,M\}$, such that at each vertex of degree at least three, the sum of the labels on the in-coming edges is equal to the sum of the labels on the out-going edges. A graph is conservative if $M=\left|E\left(G\right)\right|$. It is worth noting that determining whether certain biregular graphs are conservative is equivalent to find integer Heffter arrays.In this work we show that the conservative number of a galaxy (a disjoint union of stars) of size $M$ is $M$ for $M\equiv 0$, $3(mod4)$, and $M+1$ otherwise. Consequently, given positive integers $m_1$, $m_2$, …, $m_n$ with $m_i\ge 3$ for $1\le i\le n$, we construct a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of infinitely many circulant graphs, generalizing a result of Bryant, Gavlas and Ling (2003). In particular, it allows us to construct a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of the complete graph $K_{2M+1}$, where $M={\sum }_{i=1}^n{m}_i$. Also, we prove necessary and sufficient conditions for the existence of a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of $K_{2M+2}?F$, where $F$ is a 1-factor. Furthermore, we give a sufficient condition for a subset of ${\mathbb{Z}}_v?\left\{0\right\}$ to be sequenceable.     

List colouring of graphs and generalized Dyck paths

The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a connection between the Catalan numbers and list colouring of graphs. Assume $G$ is a graph and $f:V\left(G\right)\to N$ is a mapping. For a nonnegative integer $m$, let $f^{\left(m\right)}$ be the extension of $f$ to the graph $G$

 $\overline{K_m}$ for which $f^{\left(m\right)}\left(v\right)=\left|V\left(G\right)\right|$ for each vertex $v$ of $\overline{K_m}$. Let $m_c(G,f)$ be the minimum $m$ such that $G$

 $\overline{K_m}$ is not $f^{\left(m\right)}$-choosable and $m_p(G,f)$ be the minimum $m$ such that $G$

 $\overline{K_m}$ is not $f^{\left(m\right)}$-paintable. We study the parameter $m_c(K_n,f)$ and $m_p(K_n,f)$ for arbitrary mappings $f$. For $\overrightarrow{x}=(x_1,x_2,\dots ,x_n)$, an $\overrightarrow{x}$-dominated path ending at $(a,b)$ is a monotonic path $P$ of the $a\times b$ grid from $(0,0)$ to $(a,b)$ such that each vertex $(i,j)$ on $P$ satisfies $i\le x_{j+1}$. Let $\psi \left(\overrightarrow{x}\right)$ be the number of $\overrightarrow{x}$-dominated paths ending at $(x_n,n)$. By this definition, the Catalan number $C_n$ equals $\psi \left((0,1,\dots ,n?1)\right)$. This paper proves that if $G=K_n$ has vertices $v_1,v_2,\dots ,v_n$ and $f\left(v_1\right)\le f\left(v_2\right)\le \dots \le f\left(v_n\right)$, then $m_c(G,f)=m_p(G,f)=\psi \left(\overrightarrow{x}\left(f\right)\right)$, where $\overrightarrow{x}\left(f\right)=(x_1,x_2,\dots ,x_n)$ and $x_i=f\left(v_i\right)?i$ for $i=1,2,\dots ,n$. Therefore, if $f\left(v_i\right)=n$, then $m_c(K_n,f)=m_p(K_n,f)$ equals the Catalan number $C_n$. We also show that if $G=G_1\cup G_2\cup ?\cup G_p$ is the disjoint union of graphs $G_1,G_2,\dots ,G_p$ and $f=f_1\cup f_2\cup ?\cup f_p$, then $m_c(G,f)={\prod }_{i=1}^p{m}_c(G_i,f_i)$ and $m_p(G,f)={\prod }_{i=1}^p{m}_p(G_i,f_i)$. This generalizes a result in Carraher et al. (2014), where the case each $G_i$ is a copy of $K_1$ is considered.     

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 $G_2=K_1\vee (K_1+2K_2).$