On the size of special class 1 graphs and (P3;k)-co-critical graphs

A well-known theorem of Vizing states that if G is a simple graph with maximum degree Δ, then the chromatic index ${\chi }^{\prime }(G)$ of G is Δ or $\mathrm{\Delta }+1$. A graph G is class 1 if ${\chi }^{\prime }(G)=\mathrm{\Delta }$, and class 2 if ${\chi }^{\prime }(G)=\mathrm{\Delta }+1$; G is Δ-critical if it is connected, class 2 and ${\chi }^{\prime }(G-e)<{\chi }^{\prime }(G)$ for every $e\in E(G)$. A long-standing conjecture of Vizing from 1968 states that every Δ-critical graph on n vertices has at least $(n(\mathrm{\Delta }-1)+3)/2$ edges. We initiate the study of determining the minimum number of edges of class 1 graphs G, in addition, ${\chi }^{\prime }(G+e)={\chi }^{\prime }(G)+1$ for every $e\in E(\bar{G})$. Such graphs have intimate relation to $(P_3;k)$-co-critical graphs, where a non-complete graph G is $(P_3;k)$-co-critical if there exists a k-coloring of $E(G)$ such that G does not contain a monochromatic copy of $P_3$ but every k-coloring of $E(G+e)$ contains a monochromatic copy of $P_3$ for every $e\in E(\bar{G})$. We use the bound on the size of the aforementioned class 1 graphs to study the minimum number of edges over all $(P_3;k)$-co-critical graphs. We prove that if G is a $(P_3;k)$-co-critical graph on $n\ge k+2$ vertices, then$e(G)\ge \frac{k}{2}(n-\lceil \frac{k}{2}\rceil -\epsilon )+\left(\begin{array}{c}\lceil k/2\rceil +\epsilon \\ 2\end{array}\right),$ where ε is the remainder of $n-\lceil k/2\rceil$ when divided by 2. This bound is best possible for all $k\ge 1$ and $n\ge \lceil 3k/2\rceil +2$.     

Classification of metabelian 2-groups G with Gab ≃ (2,2n),n ≥ 2, and rank d(G′)=2; Applications to real quadratic number fields

We characterize all finite metabelian 2-groups G whose abelianizations $G^{\mathrm{ab}}$ are of type $(2,2^n)$, with $n\ge 2$, and for which their commutator subgroups $G^{\prime }$ have $\text{rank}=2$. This is given in terms of the order of the abelianizations of the maximal subgroups and the structure of the abelianizations of those normal subgroups of index 4 in G. We then translate these group theoretic properties to give a characterization of number fields k with 2-class group ${\mathrm{Cl}}_2(k)?(2,2^n)$, $n\ge 2$, such that the rank of ${\mathrm{Cl}}_2(k^1)=2$ where $k^1$ is the Hilbert 2-class field of k. In particular, we apply all this to real quadratic number fields whose discriminants are a sum of two squares.