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 $\frac{4}{9}\left|V\left(G_n\right)\right|$.     

Problems on matchings and independent sets of a graph

Let $G$ be a finite simple graph. For $X?V\left(G\right)$, the difference of $X$, $d\left(X\right)?\left|X\right|?\left|N\left(X\right)\right|$ where $N\left(X\right)$ is the neighborhood of $X$ and $max\{d\left(X\right):X?V\left(G\right)\}$ is called the critical difference of $G$. $X$ is called a critical set if $d\left(X\right)$ equals the critical difference and $\ker \left(G\right)$ is the intersection of all critical sets. $\mathrm{diadem}\left(G\right)$ is the union of all critical independent sets. An independent set $S$ is an inclusion minimal set with$d\left(S\right)>0$ if no proper subset of $S$ has positive difference.A graph $G$ is called a König–Egerváry graph if the sum of its independence number $\alpha \left(G\right)$ and matching number $\mu \left(G\right)$ equals $\left|V\left(G\right)\right|$.In this paper, we prove a conjecture which states that for any graph the number of inclusion minimal independent set $S$ with $d\left(S\right)>0$ is at least the critical difference of the graph.We also give a new short proof of the inequality $\left|\ker \left(G\right)\right|+\left|\mathrm{diadem}\left(G\right)\right|\le 2\alpha \left(G\right)$.A characterization of unicyclic non-König–Egerváry graphs is also presented and a conjecture which states that for such a graph $G$, the critical difference equals $\alpha \left(G\right)?\mu \left(G\right)$, is proved.We also make an observation about $\ker \left(G\right)$ using Edmonds–Gallai Structure Theorem as a concluding remark.     

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.     

Generalizing Tutte’s theorem and maximal non-matchable graphs

A graph $G$ has a perfect matching if and only if $0$ is not a root of its matching polynomial $\mu (G,x)$. Thus, Tutte’s famous theorem asserts that $0$ is not a root of $\mu (G,x)$ if and only if $c_{\mathrm{odd}}(G?S)\le \left|S\right|$ for all $S?V\left(G\right)$, where $c_{\mathrm{odd}}\left(G\right)$ denotes the number of odd components of $G$. Tutte’s theorem can be proved using a characterization of the structure of maximal non-matchable graphs, that is, the edge-maximal graphs among those having no perfect matching. In this paper, we prove a generalized version of Tutte’s theorem in terms of avoiding any given real number $\theta$ as a root of $\mu (G,x)$. We also extend maximal non-matchable graphs to maximal $\theta$-non-matchable graphs and determine the structure of such graphs.     

Maximal linear groups induced on the Frattini quotient of a p-group

Let $p>3$ be a prime. For each maximal subgroup $H?\mathrm{GL}(d,p)$ with $|H|?p^{3d+1}$, we construct a d-generator finite p-group G with the property that $\mathrm{Aut}(G)$ induces H on the Frattini quotient $G/\mathrm{\Phi }(G)$ and $|G|?p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on $G/\mathrm{\Phi }(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.     

On disjoint matchings in cubic graphs

For $i=2,3$ and a cubic graph $G$ let ${\nu }_i\left(G\right)$ denote the maximum number of edges that can be covered by $i$ matchings. We show that ${\nu }_2\left(G\right)\ge \frac{4}{5}\left|V\left(G\right)\right|$ and ${\nu }_3\left(G\right)\ge \frac{7}{6}\left|V\left(G\right)\right|$. Moreover, it turns out that ${\nu }_2\left(G\right)\le \frac{\left|V\left(G\right)\right|+2{\nu }_3\left(G\right)}{4}$.     

A new formula for the decycling number of regular graphs

The decycling number $?\left(G\right)$ of a graph $G$ is the smallest number of vertices which can be removed from $G$ so that the resultant graph contains no cycle. A decycling set containing exactly $?\left(G\right)$ vertices of $G$ is called a $?$-set. For any decycling set $S$ of a $k$-regular graph $G$, we show that $\left|S\right|=\frac{\beta \left(G\right)+m\left(S\right)}{k?1}$, where $\beta \left(G\right)$ is the cycle rank of $G$, $m\left(S\right)=c+\left|E\left(S\right)\right|?1$ is the margin number of $S$, $c$ and $\left|E\left(S\right)\right|$ are, respectively, the number of components of $G?S$ and the number of edges in $G\left[S\right]$. In particular, for any $?$-set $S$ of a 3-regular graph $G$, we prove that $m\left(S\right)=\xi \left(G\right)$, where $\xi \left(G\right)$ is the Betti deficiency of $G$. This implies that the decycling number of a 3-regular graph $G$ is $\frac{\beta \left(G\right)+\xi \left(G\right)}{2}$. Hence $?\left(G\right)=?\frac{\beta \left(G\right)}{2}?$ for a 3-regular upper-embeddable graph $G$, which concludes the results in [Gao et al., 2015, Wei and Li, 2013] and solves two open problems posed by Bau and Beineke (2002). Considering an algorithm by Furst et al., (1988), there exists a polynomial time algorithm to compute $Z\left(G\right)$, the cardinality of a maximum nonseparating independent set in a $3$-regular graph $G$, which solves an open problem raised by Speckenmeyer (1988). As for a 4-regular graph $G$, we show that for any $?$-set $S$ of $G$, there exists a spanning tree $T$ of $G$ such that the elements of $S$ are simply the leaves of $T$ with at most two exceptions providing $?\left(G\right)=?\frac{\beta \left(G\right)}{3}?$. On the other hand, if $G$ is a loopless graph on $n$ vertices with maximum degree at most $4$, then

The above two upper bounds are tight, and this makes an extension of a result due to Punnim (2006).     

On the edge dimension of a graph

Given a connected graph $G(V,E)$, the edge dimension, denoted $\mathrm{edim}\left(G\right)$, is the least size of a set $S?V$ that distinguishes every pair of edges of $G$, in the sense that the edges have pairwise different tuples of distances to the vertices of $S$. The notation was introduced by Kelenc, Tratnik, and Yero, and in their paper they posed several questions about various properties of $\mathrm{edim}$. In this article we answer two of these questions: we classify the graphs on $n$ vertices for which $\mathrm{edim}\left(G\right)=n?1$ and show that $\frac{\mathrm{edim}\left(G\right)}{dim\left(G\right)}$ is not bounded from above (here $dim\left(G\right)$ is the standard metric dimension of $G$). We also compute $\mathrm{edim}(G\square P_m)$ and $\mathrm{edim}(G+K_1)$.     

Equitable Δ-coloring of graphs

Consider a graph $G$ consisting of a vertex set $V\left(G\right)$ and an edge set $E\left(G\right)$. Let $\Delta \left(G\right)$ and $\chi \left(G\right)$ denote the maximum degree and the chromatic number of $G$, respectively. We say that $G$ is equitably $\Delta \left(G\right)$-colorable if there exists a proper $\Delta \left(G\right)$-coloring of $G$ such that the sizes of any two color classes differ by at most one. Obviously, if $G$ is equitably $\Delta \left(G\right)$-colorable, then $\Delta \left(G\right)\ge \chi \left(G\right)$. Conversely, even if $G$ satisfies $\Delta \left(G\right)\ge \chi \left(G\right)$, we cannot guarantee that $G$ must be equitably $\Delta \left(G\right)$-colorable. In 1994, the Equitable $\Delta$-Coloring Conjecture $\left(E\Delta \mathrm{CC}\right)$ asserts that a connected graph $G$ with $\Delta \left(G\right)\ge \chi \left(G\right)$ is equitably $\Delta \left(G\right)$-colorable if $G$ is different from $K_{2n+1,2n+1}$ for all $n\ge 1$. In this paper, we give necessary conditions for a graph $G$ (not necessarily connected) with $\Delta \left(G\right)\ge \chi \left(G\right)$ to be equitably $\Delta \left(G\right)$-colorable and prove that those necessary conditions are also sufficient conditions when $G$ is a bipartite graph, or $G$ satisfies $\Delta \left(G\right)\ge \frac{\left|V\left(G\right)\right|}{3}+1$, or $G$ satisfies $\Delta \left(G\right)\le 3$.     

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 $G$.