On prisms,Möbius ladders and the cycle space of dense graphs

For a graph $G$, let $\left|G\right|$ denote its number of vertices, $\delta \left(G\right)$ its minimum degree and $Z_1(G;{\mathbb{F}}_2)$ its cycle space. Call a graph Hamilton-generated if and only if every cycle in $G$ is a symmetric difference of some Hamilton circuits of $G$. The main purpose of this paper is to prove: for every $\gamma >0$ there exists $n_0\in \mathbb{Z}$ such that for every graph $G$ with $\left|G\right|⩾n_0$ vertices,

• (1)if $\delta \left(G\right)⩾(\frac{1}{2}+\gamma )\left|G\right|$ and $\left|G\right|$ is odd, then $G$ is Hamilton-generated,
• (2)if $\delta \left(G\right)⩾(\frac{1}{2}+\gamma )\left|G\right|$ and $\left|G\right|$ is even, then the set of all Hamilton circuits of $G$ generates a codimension-one subspace of $Z_1(G;{\mathbb{F}}_2)$ and the set of all circuits of $G$ having length either $\left|G\right|-1$ or $\left|G\right|$ generates all of $Z_1(G;{\mathbb{F}}_2)$,
• (3)if $\delta \left(G\right)⩾(\frac{1}{4}+\gamma )\left|G\right|$ and $G$ is balanced bipartite, then $G$ is Hamilton-generated.

All these degree-conditions are essentially best-possible. The implications in (1) and (2) give an asymptotic affirmative answer to a special case of an open conjecture which according to [I.B.-A. Hartman, Long cycles generate the cycle space of a graph, European J. Combin. 4 (3) (1983) 237–246] originates with A. Bondy.     

Proper connection and size of graphs

An edge-coloured graph $G$ is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a connected graph $G,$ denoted by $\mathrm{pc}\left(G\right)$, is the smallest number of colours that are needed in order to make $G$ properly connected. Our main result is the following: Let $G$ be a connected graph of order $n$ and $k\ge 2$. If $\left|E\left(G\right)\right|\ge \left(\genfrac{}{}{0ex}{}{n?k?1}{2}\right)+k+2$, then $\mathrm{pc}\left(G\right)\le k$ except when $k=2$ and $G\in \{G_1,G_2\},$ where $G_1=K_1\vee (2K_1+K_2)$ and $G_2=K_1\vee (K_1+2K_2).$     

Steiner tree packing number and tree connectivity

Let $S$ be a set of at least two vertices in a graph $G$. A subtree $T$ of $G$ is a $S$-Steiner tree if $S?V\left(T\right)$. Two $S$-Steiner trees $T_1$ and $T_2$ are edge-disjoint (resp. internally vertex-disjoint) if $E\left(T_1\right)\cap E\left(T_2\right)=?$ (resp. $E\left(T_1\right)\cap E\left(T_2\right)=?$ and $V\left(T_1\right)\cap V\left(T_2\right)=S$). Let ${\lambda }_G\left(S\right)$ (resp. ${\kappa }_G\left(S\right)$) be the maximum number of edge-disjoint (resp. internally vertex-disjoint) $S$-Steiner trees in $G$, and let ${\lambda }_k\left(G\right)$ (resp. ${\kappa }_k\left(G\right)$) be the minimum ${\lambda }_G\left(S\right)$ (resp. ${\kappa }_G\left(S\right)$) for $S$ ranges over all $k$-subset of $V\left(G\right)$. Kriesell conjectured that if ${\lambda }_G\left(\{x,y\}\right)\ge 2k$ for any $x,y\in S$, then ${\lambda }_G\left(S\right)\ge k$. He proved that the conjecture holds for $\left|S\right|=3,4$. In this paper, we give a short proof of Kriesell’s Conjecture for $\left|S\right|=3,4$, and also show that ${\lambda }_k\left(G\right)\ge ?\frac{1}{k?1}?\frac{k?}{2}??$ (resp. ${\kappa }_k\left(G\right)\ge ?\frac{1}{k?1}?\frac{k?}{2}??$ ) if $\lambda \left(G\right)\ge ?$ (resp. $\kappa \left(G\right)\ge ?$) in $G$, where $k=3,4$. Moreover, we also study the relation between ${\kappa }_k\left(L\left(G\right)\right)$ and ${\lambda }_k\left(G\right)$, where $L\left(G\right)$ is the line graph of $G$.     

On a conjecture of Gentner and Rautenbach

Gentner and Rautenbach conjectured that the size of a minimum zero forcing set in a connected graph on $n$ vertices with maximum degree $3$ is at most $\frac{1}{3}n+2$. We disprove this conjecture by constructing a collection of connected graphs $\left\{G_n\right\}$ with maximum degree 3 of arbitrarily large order having zero forcing number at least $\frac{4}{9}\left|V\left(G_n\right)\right|$.     

Extremal product-one free sequences in Dihedral and Dicyclic Groups

Let $G$ be a finite group, written multiplicatively. The Davenport constant of $G$ is the smallest positive integer $D\left(G\right)$ such that every sequence of $G$ with $D\left(G\right)$ elements has a non-empty subsequence with product $1$. Let $D_{2n}$ be the Dihedral Group of order $2n$ and $Q_{4n}$ be the Dicyclic Group of order $4n$. Zhuang and Gao (2005) showed that $D\left(D_{2n}\right)=n+1$ and Bass (2007) showed that $D\left(Q_{4n}\right)=2n+1$. In this paper, we give explicit characterizations of all sequences $S$ of $G$ such that $\left|S\right|=D\left(G\right)?1$ and $S$ is free of subsequences whose product is 1, where $G$ is equal to $D_{2n}$ or $Q_{4n}$ for some $n$.     

Maximum average degree and relaxed coloring

We say a graph is $(d,d,\dots ,d,0,\dots ,0)$-colorable with $a$ of $d$’s and $b$ of $0$’s if $V\left(G\right)$ may be partitioned into $b$ independent sets $O_1,O_2,\dots ,O_b$ and $a$ sets $D_1,D_2,\dots ,D_a$ whose induced graphs have maximum degree at most $d$. The maximum average degree, $mad\left(G\right)$, of a graph $G$ is the maximum average degree over all subgraphs of $G$. In this note, for nonnegative integers $a,b$, we show that if $mad\left(G\right)<\frac{4}{3}a+b$, then $G$ is $(1_1,1_2,\dots ,1_a,0_1,\dots ,0_b)$-colorable.     

Cycles with a chord in dense graphs

A cycle of order $k$ is called a $k$-cycle. A non-induced cycle is called a chorded cycle. Let $n$ be an integer with $n\ge 4$. Then a graph $G$ of order $n$ is chorded pancyclic if $G$ contains a chorded $k$-cycle for every integer $k$ with $4\le k\le n$. Cream, Gould and Hirohata (Australas. J. Combin. 67 (2017), 463–469) proved that a graph of order $n$ satisfying ${deg}_{G}u+{deg}_{G}v\ge n$ for every pair of nonadjacent vertices $u$,  $v$ in $G$ is chorded pancyclic unless $G$ is either $K_{\frac{n}{2},\frac{n}{2}}$ or $K_3\square K_2$, the Cartesian product of $K_3$ and $K_2$. They also conjectured that if $G$ is Hamiltonian, we can replace the degree sum condition with the weaker density condition $\left|E\left(G\right)\right|\ge \frac{1}{4}{n}^2$ and still guarantee the same conclusion. In this paper, we prove this conjecture by showing that if a graph $G$ of order $n$ with $\left|E\left(G\right)\right|\ge \frac{1}{4}{n}^2$ contains a $k$-cycle, then $G$ contains a chorded $k$-cycle, unless $k=4$ and $G$ is either $K_{\frac{n}{2},\frac{n}{2}}$ or $K_3\square K_2$, Then observing that $K_{\frac{n}{2},\frac{n}{2}}$ and $K_3\square K_2$ are exceptions only for $k=4$, we further relax the density condition for sufficiently large $k$.