1. CMB 2017 (vol 60 pp. 319)
 Khojasteh, Sohiela; Nikmehr, Mohammad Javad

The Weakly Nilpotent Graph of a Commutative Ring
Let $R$ be a commutative ring with nonzero identity. In this
paper, we introduced the weakly nilpotent graph of a commutative
ring. The weakly nilpotent graph of $R$ is denoted by $\Gamma_w(R)$
is a graph with the vertex set $R^{*}$ and two vertices $x$ and
$y$ are adjacent if and only if $xy\in N(R)^{*}$, where $R^{*}=R\setminus\{0\}$
and $N(R)^{*}$ is the set of all nonzero nilpotent elements
of $R$. In this article, we determine the diameter of weakly
nilpotent graph of an Artinian ring. We prove that if $\Gamma_w(R)$
is a forest, then $\Gamma_w(R)$ is a union of a star and some
isolated vertices. We study the clique number, the chromatic
number and the independence number of $\Gamma_w(R)$. Among other
results, we show that for an Artinian ring $R$, $\Gamma_w(R)$
is not a disjoint union of cycles or a unicyclic graph. For Artinan
ring, we determine $\operatorname{diam}(\overline{\Gamma_w(R)})$. Finally, we
characterize all commutative rings $R$ for which $\overline{\Gamma_w(R)}$
is a cycle, where $\overline{\Gamma_w(R)}$ is the complement
of the weakly nilpotent graph of $R$.
Keywords:weakly nilpotent graph, zerodivisor graph, diameter, girth Categories:05C15, 16N40, 16P20 

2. CMB 2016 (vol 60 pp. 154)
 Liu, Ye

On Chromatic Functors and Stable Partitions of Graphs
The chromatic functor of a simple graph is a functorization of
the chromatic polynomial. M. Yoshinaga showed
that two finite graphs have isomorphic chromatic functors if
and only if they have the same chromatic polynomial. The key
ingredient in the proof is the use of stable partitions of graphs.
The latter is shown to be closely related to chromatic functors.
In this note, we further investigate some interesting properties
of chromatic functors associated to simple graphs using stable
partitions. Our first result is the determination of the group
of natural automorphisms of the chromatic functor, which is in
general a larger group than the automorphism group of the graph.
The second result is that the composition of the chromatic functor
associated to a finite graph restricted to the category $\mathrm{FI}$
of finite sets and injections with the free functor into the
category of complex vector spaces yields a consistent sequence
of representations of symmetric groups which is representation
stable in the sense of ChurchFarb.
Keywords:chromatic functor, stable partition, representation stability Categories:05C15, 20C30 

3. CMB 2016 (vol 59 pp. 641)
 Shaveisi, Farzad

Some Results on the Annihilatingideal Graphs
The annihilatingideal graph
of a commutative ring $R$, denoted by $\mathbb{AG}(R)$, is a
graph whose vertex set consists of all nonzero annihilating
ideals and two distinct
vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. Here,
we show that if $R$ is a reduced ring and the independence
number of $\mathbb{AG}(R)$ is finite, then the edge chromatic
number of $\mathbb{AG}(R)$ equals its maximum degree
and this number equals $2^{{\rm Min}(R)1}1$; also, it is
proved that the independence number of $\mathbb{AG}(R)$ equals
$2^{{\rm Min}(R)1}$, where ${\rm Min}(R)$ denotes the set
of minimal prime ideals of $R$.
Then we give some criteria for a graph to be isomorphic with
an annihilatingideal graph of a ring.
For example, it is shown that every bipartite annihilatingideal
graph is a complete bipartite graph with at most two horns. Among
other results, it is shown that a finite graph $\mathbb{AG}(R)$
is not Eulerian, and it is Hamiltonian if and only if $R$ contains
no Gorenstain ring as its direct summand.
Keywords:annihilatingideal graph, independence number, edge chromatic number, bipartite, cycle Categories:05C15, 05C69, 13E05, 13E10 

4. CMB 2016 (vol 59 pp. 440)
 Zhang, Haihui

A Note on 3choosability of Planar Graphs Related to Montanssier's Conjecture
A graph $G=(V,E)$ is $L$colorable if for a given list
assignment $L=\{L(v):v\in V(G)\}$, there exists a proper coloring
$c$ of $G$ such that $c(v)\in L(v)$ for all $v\in V$. If $G$ is
$L$colorable for every list assignment $L$ with $L(v)\geq
k$ for
all $v\in V$, then $G$ is said to be $k$choosable. Montassier
(Inform. Process. Lett. 99 (2006) 6871) conjectured that every
planar
graph without cycles of length 4, 5, 6, is 3choosable. In this
paper,
we prove that every planar graph without 5, 6 and 10cycles,
and
without two triangles at distance less than 3 is 3choosable.
Keywords:choosability, planar graph, cycle Category:05C15 

5. CMB 2015 (vol 59 pp. 190)
 Raeisi, Ghaffar; Zaghian, Ali

Ramsey Number of Wheels Versus Cycles and Trees
Let $G_1, G_2, \dots , G_t$ be arbitrary graphs. The
Ramsey number $R(G_1, G_2, \dots, G_t)$ is the smallest positive
integer $n$ such that if the edges of the complete graph $K_n$
are
partitioned into $t$ disjoint color classes giving $t$ graphs
$H_1,H_2,\dots,H_t$, then at least one $H_i$ has a subgraph
isomorphic to $G_i$. In this paper, we provide the exact value
of
the $R(T_n,W_m)$ for odd $m$, $n\geq m1$, where $T_n$ is
either a caterpillar, a tree with diameter at most four or a
tree
with a vertex adjacent to at least $\lceil
\frac{n}{2}\rceil2$ leaves. Also, we
determine $R(C_n,W_m)$ for even integers $n$ and $m$, $n\geq
m+500$, which improves a result of Shi and confirms a
conjecture of Surahmat et al. In addition, the multicolor Ramsey
number of trees
versus an odd wheel is discussed in this paper.
Keywords:Ramsey number, wheel, tree, cycle Categories:05C15, 05C55, 05C65 

6. CMB 2015 (vol 58 pp. 317)
7. CMB 2013 (vol 56 pp. 449)
 Akbari, S.; Chavooshi, M.; Ghanbari, M.; Zare, S.

The $f$Chromatic Index of a Graph Whose $f$Core has Maximum Degree $2$
Let $G$ be a graph. The minimum number of colors needed to color the edges of
$G$ is called the chromatic index of $G$ and is denoted by $\chi'(G)$.
It is wellknown that $\Delta(G) \leq \chi'(G) \leq \Delta(G)+1$, for any
graph $G$, where $\Delta(G)$ denotes the maximum degree of $G$. A graph $G$ is said to be
Class $1$ if $\chi'(G) = \Delta(G)$ and Class $2$ if
$\chi'(G) = \Delta(G) + 1$. Also, $G_\Delta$ is the induced subgraph on all vertices of degree $\Delta(G)$.
Let $f:V(G)\rightarrow \mathbb{N}$ be a function.
An $f$coloring of a graph $G$ is a coloring of the edges
of $E(G)$ such that each color appears at each vertex $v\in V(G)$ at
most $f (v)$ times. The minimum number of colors needed
to $f$color $G$ is called the $f$chromatic index of $G$ and
is denoted by $\chi'_{f}(G)$. It was shown that for every graph $G$, $\Delta_{f}(G)\le \chi'_{f}(G)\le \Delta_{f}(G)+1$, where $\Delta_{f}(G)=\max_{v\in V(G)} \big\lceil \frac{d_G(v)}{f(v)}\big\rceil$. A graph $G$ is said to be $f$Class $1$ if $\chi'_{f}(G)=\Delta_{f}(G)$, and $f$Class $2$, otherwise. Also, $G_{\Delta_f}$ is the induced subgraph of $G$ on $\{v\in V(G):\,\frac{d_G(v)}{f(v)}=\Delta_{f}(G)\}$.
Hilton and Zhao showed that if $G_{\Delta}$ has maximum degree two and $G$ is Class $2$, then $G$ is critical, $G_{\Delta}$ is a disjoint union of cycles and $\delta(G)=\Delta(G)1$, where $\delta(G)$ denotes the minimum degree of $G$, respectively. In this paper, we generalize this theorem to $f$coloring of graphs. Also, we determine the $f$chromatic index of a connected graph $G$ with $G_{\Delta_f}\le 4$.
Keywords:$f$coloring, $f$Core, $f$Class $1$ Categories:05C15, 05C38 

8. CMB 2009 (vol 52 pp. 451)
 Pach, János; Tardos, Gábor; Tóth, Géza

Indecomposable Coverings
We prove that for every $k>1$, there exist $k$fold coverings of the
plane (i) with strips, (ii) with axisparallel rectangles, and
(iii) with homothets of any fixed concave quadrilateral, that cannot
be decomposed into two coverings. We also construct for every
$k>1$ a set of points $P$ and a family of disks $\cal D$ in the
plane, each containing at least $k$ elements of $P$, such that, no
matter how we color the points of $P$ with two colors,
there
exists a disk $D\in{\cal D}$ all of whose points are of the same
color.
Categories:52C15, 05C15 

9. CMB 2000 (vol 43 pp. 108)
 Sanders, Daniel P.; Zhao, Yue

On the Entire Coloring Conjecture
The Four Color Theorem says that the faces (or vertices) of a plane
graph may be colored with four colors. Vizing's Theorem says that the
edges of a graph with maximum degree $\Delta$ may be colored with
$\Delta+1$ colors. In 1972, Kronk and Mitchem conjectured that the
vertices, edges, and faces of a plane graph may be simultaneously
colored with $\Delta+4$ colors. In this article, we give a simple
proof that the conjecture is true if $\Delta \geq 6$.
Categories:05C15, 05C10 
