Subgraph densities in a surface

Given a fixed graph H that embeds in a surface $\Sigma$ , what is the maximum number of copies of H in an n-vertex graph G that embeds in $\Sigma$ ? We show that the answer is $\Theta(n^{f(H)})$ , where f(H) is a graph invariant called the ‘flap-number’ of H, which is independent of $\Sigma$ . This simultaneously answers two open problems posed by Eppstein ((1993) J. Graph Theory17(3) 409–416.). The same proof also answers the question for minor-closed classes. That is, if H is a $K_{3,t}$ minor-free graph, then the maximum number of copies of H in an n-vertex $K_{3,t}$ minor-free graph G is $\Theta(n^{f'(H)})$ , where f′(H) is a graph invariant closely related to the flap-number of H. Finally, when H is a complete graph we give more precise answers.

Bilangan Terhubung Titik Pelangi pada Graf Hasil Operasi Korona Graf Prisma (P_(m,2)) dan Graf Lintasan (P_3)

Rainbow vertex-connection number is the minimum k-coloring on the vertex graph G and is denoted by rvc(G). Besides, the rainbow-vertex connection number can be applied to some special graphs, such as prism graph and path graph. Graph operation is a method used to create a new graph by combining two graphs. Therefore, this research uses corona product operation to form rainbow-vertex connection number at the graph resulting from corona product operation of prism graph and path graph (Pm,2 P3) (P3 Pm,2). The results of this study obtain that the theorem of rainbow vertex-connection number at the graph resulting from corona product operation of prism graph and path graph (Pm,2 P3) (P3 Pm,2) for 3 = m = 7 are rvc (G) = 2m rvc (G) = 2.

Smaller Extended Formulations for Spanning Tree Polytopes in Minor-closed Classes and Beyond

Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following from the work of Wong (1980) and Martin (1991). It also extends a result of Fiorini, Huynh, Joret, and Pashkovich (2017), who obtained a $O(n^{3/2})$ bound for graphs embedded in a fixed surface. Our proof works more generally for all graph classes admitting strongly sublinear balanced separators: We prove that for every constant $\beta$ with $0<\beta<1$, if $\mathcal G$ is a graph class closed under induced subgraphs such that all $n$-vertex graphs in $\mathcal G$ have balanced separators of size $O(n^\beta)$, then the extension complexity of the spanning tree polytope of every connected $n$-vertex graph in $\mathcal{G}$ is $O(n^{1+\beta})$. We in fact give two proofs of this result, one is a direct construction of the extended formulation, the other is via communication protocols. Using the latter approach we also give a short proof of the $O(n)$ bound for planar graphs due to Williams (2002).

A spanning bandwidth theorem in random graphs

The bandwidth theorem of Böttcher, Schacht and Taraz states that any n-vertex graph G with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). Recently, a subset of the authors proved a random graph analogue of this statement: for $p\gg \big(\tfrac{\log n}{n}\big)^{1/\Delta}$ a.a.s. each spanning subgraph G of G(n,p) with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all n-vertex k-colourable graphs H with maximum degree $\Delta$ , bandwidth o(n), and at least $C p^{-2}$ vertices not contained in any triangle. This restriction on vertices in triangles is necessary, but limiting. In this paper, we consider how it can be avoided. A special case of our main result is that, under the same conditions, if additionally all vertex neighbourhoods in G contain many copies of $K_\Delta$ then we can drop the restriction on H that $Cp^{-2}$ vertices should not be in triangles.

Generalized Planar Turán Numbers

In a generalized Turán problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $n$-vertex graph not containing $F$ as a subgraph. We consider generalized Turán problems where the host graph is planar. In particular, we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most $2\ell$, for all $\ell$, $\ell \geqslant 1$. We also determine the order of magnitude of the maximum number of cycles of a given length in a planar $C_4$-free graph. An exact result is given for the maximum number of $5$-cycles in a $C_4$-free planar graph. Multiple conjectures are also introduced.

Practical Compact Routing on Sparse Graphs

We describe a simple and practical algorithm for compact routing on graphs which admit compact and balanced vertex separators. Using a recursive nested dissection of then-vertex graph based on these separators, we construct routing tables with as few as O(log n) entries per vertex in a preprocessing step. They support handshaking-based routing on the graph with moderate stretch, where the handshaking can be implemented similarly to a DNS lookup. We describe a basic version of the algorithm that requires modifiable headers and a more advanced version which eliminates this need and provides better stretch. A number of algorithmic parameters control a graceful tradeoff between the size of the routing tables and the stretch. Our routing algorithm is most effective on planar graphs and unit disk graphs of moderate edge/vertex density.

Practical Compact Routing on Sparse Graphs

We describe a simple and practical algorithm for compact routing on graphs which admit compact and balanced vertex separators. Using a recursive nested dissection of then-vertex graph based on these separators, we construct routing tables with as few as O(log n) entries per vertex in a preprocessing step. They support handshaking-based routing on the graph with moderate stretch, where the handshaking can be implemented similarly to a DNS lookup. We describe a basic version of the algorithm that requires modifiable headers and a more advanced version which eliminates this need and provides better stretch. A number of algorithmic parameters control a graceful tradeoff between the size of the routing tables and the stretch. Our routing algorithm is most effective on planar graphs and unit disk graphs of moderate edge/vertex density.

On Problem of Finding all Maximal Induced Bicliques of Hypergraph

The problem of finding all maximal induced bicliques of a hypergraph is considered in this paper. A theorem about connection between induced bicliques of the hypergraph H and corresponding vertex graph L2(H) is proved. An algorithm for finding all maximal induced bicliques is proposed and computational experiments with its applying are presented