Flow number and circular flow number of signed cubic graphs

Let $\mathrm{\Phi }(G,\sigma )$ and ${\mathrm{\Phi }}_c(G,\sigma )$ denote the flow number and the circular flow number of a flow-admissible signed graph $(G,\sigma )$, respectively. It is known that $\mathrm{\Phi }(G)=?{\mathrm{\Phi }}_c(G)?$ for every unsigned graph G. Based on this fact, in 2011 Raspaud and Zhu conjectured that $\mathrm{\Phi }(G,\sigma )?{\mathrm{\Phi }}_c(G,\sigma )<1$ holds also for every flow-admissible signed graph $(G,\sigma )$. This conjecture was disproved by Schubert and Steffen using graphs with bridges and vertices of large degree. In this paper we focus on cubic graphs, since they play a crucial role in many open problems in graph theory. For cubic graphs we show that $\mathrm{\Phi }(G,\sigma )=3$ if and only if ${\mathrm{\Phi }}_c(G,\sigma )=3$ and if $\mathrm{\Phi }(G,\sigma )\in \{4,5\}$, then $4\le {\mathrm{\Phi }}_c(G,\sigma )\le \mathrm{\Phi }(G,\sigma )$. We also prove that all pairs of flow number and circular flow number that fulfil these conditions can be achieved in the family of bridgeless cubic graphs and thereby disprove the conjecture of Raspaud and Zhu even for bridgeless signed cubic graphs. Finally, we prove that all currently known flow-admissible graphs without nowhere-zero 5-flow have flow number and circular flow number 6 and propose several conjectures in this area.     

3-Vertices with fewest 2-neighbors in plane graphs with no long paths of 2-vertices

Let $g(k,t)$ be the minimum integer such that every plane graph with girth g at least $g(k,t)$, minimum degree $\delta =2$ and no $(k+1)$-paths consisting of vertices of degree 2, where $k\ge 1$, has a 3-vertex with at least t neighbors of degree 2, where $1\le t\le 3$.In 2015, Jendrol' and Maceková proved $g(1,1)\le 7$. Later on, Hudák et al. established $g(1,3)=10$, Jendrol', Maceková, Montassier, and Soták proved $g(1,1)\ge 7$, $g(1,2)=8$ and $g(2,2)\ge 11$, and we recently proved that $g(2,2)=11$ and $g(2,3)=14$.Thus $g(k,t)$ is already known for $k=1$ and all t. In this paper, we prove that $g(k,1)=3k+4$, $g(k,2)=3k+5$, and $g(k,3)=3k+8$ whenever $k\ge 2$.     

Maximize the Q-index of graphs with fixed order and size

In this paper, we consider two kinds of spectral extremal questions. The first asks which graph attains the maximum Q-index over all graphs of order n and size $m=n+k$? The second asks which graph attains the maximum Q-index over all $(p,q)$-bipartite graphs with $m=p+q+k$ edges? We solve the first question for $4\le k\le n?3$, and the second question for $?p?q\le k\le p?q$. The maximum Q-index on connected $(p,q)$-bipartite graphs is also determined for $?1\le k\le p?2$.     

On the weak 2-coloring number of planar graphs

For a graph $G=(V,E)$, a total ordering L on V, and a vertex $v\in V$, let ${\mathrm{Wcol}}_2[G,L,v]$ be the set of vertices $w\in V$ for which there is a path from v to w whose length is 0, 1 or 2 and whose L-least vertex is w. The weak 2-coloring number ${\mathrm{wcol}}_2(G)$ of G is the least k such that there is a total ordering L on V with $|{\mathrm{Wcol}}_2[G,L,v]|\le k$ for all vertices $v\in V$. We improve the known upper bound on the weak 2-coloring number of planar graphs from 28 to 23. As the weak 2-coloring number is the best known upper bound on the star list chromatic number of planar graphs, this bound is also improved.     

On the sizes of large subgraphs of the binomial random graph

We consider the binomial random graph $G(n,p)$, where p is a constant, and answer the following two questions.First, given $e(k)=p\left(\begin{array}{c}k\\ 2\end{array}\right)+O(k)$, what is the maximum k such that a.a.s. the binomial random graph $G(n,p)$ has an induced subgraph with k vertices and $e(k)$ edges? We prove that this maximum is not concentrated in any finite set (in contrast to the case of a small $e(k)$). Moreover, for every constant $C>0$, with probability bounded away from 0, the size of the concentration set is bigger than $C\sqrt{n/\ln ?n}$, and, for every ${\omega }_n\to \infty$, a.a.s. it is smaller than ${\omega }_n\sqrt{n/\ln ?n}$.Second, given $k>\epsilon n$, what is the maximum μ such that a.a.s. the set of sizes of k-vertex subgraphs of $G(n,p)$ contains a full interval of length μ? The answer is $\mu =\mathrm{\Theta }\left(\sqrt{(n?k)n\ln ?\left(\begin{array}{c}n\\ k\end{array}\right)}\right)$.     

On distance-regular Cayley graphs of generalized dicyclic groups

Let G be a generalized dicyclic group with identity 1. An inverse closed subset S of $G?\{1\}$ is called minimal if $〈S〉=G$ and there exists some $s\in S$ such that $〈S?\{s,s^{?1}\}〉\ne G$. In this paper, we characterize distance-regular Cayley graphs $\mathrm{Cay}(G,S)$ of G under the condition that S is minimal.     

The structure of the minimal free resolution of semigroup rings obtained by gluing

We construct a minimal free resolution of the semigroup ring $k[C]$ in terms of minimal resolutions of $k[A]$ and $k[B]$ when $〈C〉$ is a numerical semigroup obtained by gluing two numerical semigroups $〈A〉$ and $〈B〉$. Using our explicit construction, we compute the Betti numbers, graded Betti numbers, regularity and Hilbert series of $k[C]$, and prove that the minimal free resolution of $k[C]$ has a differential graded algebra structure provided the resolutions of $k[A]$ and $k[B]$ possess them. We discuss the consequences of our results in small embedding dimensions. Finally, we give an extension of our main result to semigroups in ${\mathbb{N}}^n$.