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Search: MSC category 05C15 ( Coloring of graphs and hypergraphs )

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1. CMB Online first

Khojasteh, Sohiela; Nikmehr, Mohammad Javad
The Weakly Nilpotent Graph of a Commutative Ring
Let $R$ be a commutative ring with non-zero 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 non-zero 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, zero-divisor 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 Church-Farb.

Keywords:chromatic functor, stable partition, representation stability
Categories:05C15, 20C30

3. CMB 2016 (vol 59 pp. 641)

Shaveisi, Farzad
Some Results on the Annihilating-ideal Graphs
The annihilating-ideal graph of a commutative ring $R$, denoted by $\mathbb{AG}(R)$, is a graph whose vertex set consists of all non-zero 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 annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal 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:annihilating-ideal 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 3-choosability 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) 68-71) conjectured that every planar graph without cycles of length 4, 5, 6, is 3-choosable. In this paper, we prove that every planar graph without 5-, 6- and 10-cycles, and without two triangles at distance less than 3 is 3-choosable.

Keywords:choosability, planar graph, cycle

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 $m$, $n\geq m-1$, where $T_n$ is either a caterpillar, a tree with diameter at most four or a tree with a vertex adjacent to at least $\lceil \frac{n}{2}\rceil-2$ 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)

Lewkowicz, Marek Kazimierz
Coloring Four-uniform Hypergraphs on Nine Vertices
Every 4-uniform hypergraph on 9 vertices with at most 25 edges has property B. This gives the answer $m_9(4)=26$ to a question raised in 1968 by Erdős.

Keywords:property B, coloring hypergraphs

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 well-known 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 axis-parallel 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

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