On defensive alliances and line graphs

Let $\Gamma$ be a simple graph of size $m$ and degree sequence $\delta_1\ge \delta_2\ge ... \ge \delta_n$. Let ${\cal L}(\Gamma)$ denotes the line graph of $\Gamma$. The aim of this paper is to study mathematical properties of the alliance number, ${a}({\cal L}(\Gamma)$, and the global alliance number, $\gamma_{a}({\cal L}(\Gamma))$, of the line graph of a simple graph. We show that $\lceil\frac{\delta_{n}+\delta_{n-1}-1}{2}\rceil \le {a}({\cal L}(\Gamma))\le \delta_1.$ In particular, if $\Gamma$ is a $\delta$-regular graph ($\delta>0$), then $a({\cal L}(\Gamma))=\delta$, and if $\Gamma$ is a $(\delta_1,\delta_2)$-semiregular bipartite graph, then $a({\cal L}(\Gamma))=\lceil \frac{\delta_1+\delta_2-1}{2} \rceil$. As a consequence of the study we compare $a({\cal L}(\Gamma))$ and ${a}(\Gamma)$, and we characterize the graphs having $a({\cal L}(\Gamma))<4$. Moreover, we show that the global-connected alliance number of ${\cal L}(\Gamma)$ is bounded by $\gamma_{ca}({\cal L}(\Gamma)) \ge \lceil\sqrt{D(\Gamma)+m-1}-1\rceil,$ where $D(\Gamma)$ denotes the diameter of $\Gamma$, and we show that the global alliance number of ${\cal L}(\Gamma)$ is bounded by $\gamma_{a}({\cal L}(\Gamma))\geq \lceil\frac{2m}{\delta_{1}+\delta_{2}+1}\rceil$. The case of strong alliances is studied by analogy

A proof of Mader's conjecture on large clique subdivisions in C4C_4-free graphs

Given any integers $s,t\geq 2$, we show there exists some $c=c(s,t)>0$ such that any $K_{s,t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $cd^{\frac{1}{2}\frac{s}{s-1}}$ vertices. In particular, when $s=2$ this resolves in a strong sense the conjecture of Mader in 1999 that every $C_4$-free graph has a subdivision of a clique with order linear in the average degree of the original graph. In general, the widely conjectured asymptotic behaviour of the extremal density of $K_{s,t}$-free graphs suggests our result is tight up to the constant $c(s,t)$.Comment: 25 pages, 1 figur

A note on Todorov surfaces

Let $S$ be a {\em Todorov surface}, {\it i.e.}, a minimal smooth surface of general type with $q=0$ and $p_g=1$ having an involution $i$ such that $S/i$ is birational to a $K3$ surface and such that the bicanonical map of $S$ is composed with $i.$ The main result of this paper is that, if $P$ is the minimal smooth model of $S/i,$ then $P$ is the minimal desingularization of a double cover of $\mathbb P^2$ ramified over two cubics. Furthermore it is also shown that, given a Todorov surface $S$, it is possible to construct Todorov surfaces $S_j$ with $K^2=1,...,K_S^2-1$ and such that $P$ is also the smooth minimal model of $S_j/i_j,$ where $i_j$ is the involution of $S_j.$ Some examples are also given, namely an example different from the examples presented by Todorov in \cite{To2}.Comment: 9 page

Note on bipartite graph tilings

Let s<t be two fixed positive integers. We study what are the minimum degree conditions for a bipartite graph G, with both color classes of size n=k(s+t), which ensure that G has a K_{s,t}-factor. Exact result for large n is given. Our result extends the work of Zhao, who determined the minimum degree threshold which guarantees that a bipartite graph has a K_{s,s}-factor.Comment: 6 pages, no figures; statement of the main theorem corrected (thanks to Andrzej Czygrinow and Louis DeBiasio); to appear in SIAM Journal on Discrete Mathematic

Rainbow Turán Problems

For a fixed graph H, we define the rainbow Turán number ex^*(n,H) to be the maximum number of edges in a graph on n vertices that has a proper edge-colouring with no rainbow H. Recall that the (ordinary) Turán number ex(n,H) is the maximum number of edges in a graph on n vertices that does not contain a copy of H. For any non-bipartite H we show that ex^*(n,H)=(1+o(1))ex(n,H), and if H is colour-critical we show that ex^{*}(n,H)=ex(n,H). When H is the complete bipartite graph K_{s,t} with s ≤ t we show ex^*(n,K_{s,t}) = O(n^{2-1/s}), which matches the known bounds for ex(n,K_{s,t}) up to a constant. We also study the rainbow Turán problem for even cycles, and in particular prove the bound ex^*(n,C_6) = O(n^{4/3}), which is of the correct order of magnitude