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Search: MSC category 05C10 ( Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] )

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1. CMB Online first

Chen, Yichao; Yin, Xuluo
The thickness of the Cartesian product of two graphs
The thickness of a graph $G$ is the minimum number of planar subgraphs whose union is $G.$ A $t$-minimal graph is a graph of thickness $t$ which contains no proper subgraph of thickness $t.$ In this paper, upper and lower bounds are obtained for the thickness, $t(G\Box H)$, of the Cartesian product of two graphs $G$ and $H$, in terms of the thickness $t(G)$ and $t(H)$. Furthermore, the thickness of the Cartesian product of two planar graphs and of a $t$-minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph $K_{4k,4k},$ the thickness of the Cartesian product of two complete bipartite graphs $K_{n,n}$ and $K_{n,n}$ is also given, for $n\neq 4k+1$.

Keywords:planar graph, thickness, Cartesian product, $t$-minimal graph, complete bipartite graph

2. CMB 2015 (vol 59 pp. 170)

Martínez-Pedroza, Eduardo
A Note on Fine Graphs and Homological Isoperimetric Inequalities
In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected $2$-complex $X$ with a linear homological isoperimetric inequality, a bound on the length of attaching maps of $2$-cells and finitely many $2$-cells adjacent to any edge must have a fine $1$-skeleton. We provide a positive answer to this question. We revisit a homological characterization of relative hyperbolicity, and show that a group $G$ is hyperbolic relative to a collection of subgroups $\mathcal P$ if and only if $G$ acts cocompactly with finite edge stabilizers on an connected $2$-dimensional cell complex with a linear homological isoperimetric inequality and $\mathcal P$ is a collection of representatives of conjugacy classes of vertex stabilizers.

Keywords:isoperimetric functions, Dehn functions, hyperbolic groups
Categories:20F67, 05C10, 20J05, 57M60

3. CMB 2011 (vol 56 pp. 265)

Chen, Yichao; Mansour, Toufik; Zou, Qian
Embedding Distributions of Generalized Fan Graphs
Total embedding distributions have been known for a few classes of graphs. Chen, Gross, and Rieper computed it for necklaces, close-end ladders and cobblestone paths. Kwak and Shim computed it for bouquets of circles and dipoles. In this paper, a splitting theorem is generalized and the embedding distributions of generalized fan graphs are obtained.

Keywords:total embedding distribution, splitting theorem, generalized fan graphs

4. CMB 2008 (vol 51 pp. 535)

Csorba, Péter
On the Simple $\Z_2$-homotopy Types of Graph Complexes and Their Simple $\Z_2$-universality
We prove that the neighborhood complex $\N(G)$, the box complex $\B(G)$, the homomorphism complex $\Hom(K_2,G)$and the Lov\'{a}sz complex $\L(G)$ have the same simple $\Z_2$-homotopy type in the sense of Whitehead. We show that these graph complexes are simple $\Z_2$-universal.

Keywords:graph complexes, simple $\Z_2$-homotopy, universality
Categories:57Q10, 05C10, 55P10

5. CMB 2001 (vol 44 pp. 370)

Weston, Anthony
On Locating Isometric $\ell_{1}^{(n)}$
Motivated by a question of Per Enflo, we develop a hypercube criterion for locating linear isometric copies of $\lone$ in an arbitrary real normed space $X$. The said criterion involves finding $2^{n}$ points in $X$ that satisfy one metric equality. This contrasts nicely to the standard classical criterion wherein one seeks $n$ points that satisfy $2^{n-1}$ metric equalities.

Keywords:normed spaces, hypercubes
Categories:46B04, 05C10, 05B99

6. 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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