Turán numbers of vertex-disjoint cliques in r-partite graphs

For two graphs $G$ and $H$, the Turán number$ex(G,H)$ is the maximum number of edges in a subgraph of $G$ that contains no copy of $H$. Chen, Li, and Tu determined the Turán numbers $ex(K_{m,n},k{K}_2)$ for all $k\ge 1$ Chen et al. (2009). In this paper we will determine the Turán numbers $ex(K_{a_1,\dots ,a_r},k{K}_r)$ for all $r\ge 3$ and $k\ge 1$.     

Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

In a previous work, it was shown how the linearized strain tensor field $\mathbf{e}:=\frac{1}{2}(\mathbf{?}{\mathit{u}}^T+\mathbf{?}\mathit{u})\in {\mathbb{L}}^2(\Omega )$ can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain $\Omega ?{\mathbb{R}}^3$, instead of the displacement vector field $\mathit{u}\in {\mathit{H}}^1(\Omega )$ in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition $\mathit{u}=\mathbf{0}$ on a portion ${\Gamma }_0$ of the boundary of Ω can be recast, again as boundary conditions on ${\Gamma }_0$, but this time expressed only in terms of the new unknown $\mathbf{e}\in {\mathbb{L}}^2(\Omega )$.     

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 $\mathrm{edim}(G+K_1)$.     

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).     

Degree Ramsey numbers for even cycles

Let $H\stackrel{s}{?}G$ denote that any $s$-coloring of $E\left(H\right)$ contains a monochromatic $G$. The degree Ramsey number of a graph $G$, denoted by $R_{\Delta }(G,s)$, is $min\{\Delta \left(H\right):H\stackrel{s}{?}G\}$. We consider degree Ramsey numbers where $G$ is a fixed even cycle. Kinnersley, Milans, and West showed that $R_{\Delta }(C_{2k},s)\ge 2s$, and Kang and Perarnau showed that $R_{\Delta }(C_4,s)=\Theta \left(s^2\right)$. Our main result is that $R_{\Delta }(C_6,s)=\Theta \left(s^{3∕2}\right)$ and $R_{\Delta }(C_{10},s)=\Theta \left(s^{5∕4}\right)$. Additionally, we substantially improve the lower bound for $R_{\Delta }(C_{2k},s)$ for general $k$.     

Turán number and decomposition number of intersecting odd cycles

Given a graph $H$, the Turán function $\mathrm{ex}(n,H)$ is the maximum number of edges in a graph on $n$ vertices that does not contain $H$ as a subgraph. Let $s,t$ be integers and let $H_{s,t}$ be a graph consisting of $s$ triangles and $t$ cycles of odd lengths at least 5 which intersect in exactly one common vertex. Erd?s et al. (1995) determined the Turán function $\mathrm{ex}(n,H_{s,0})$ and the corresponding extremal graphs. Recently, Hou et al. (2016) determined $\mathrm{ex}(n,H_{0,t})$ and the extremal graphs, where the $t$ cycles have the same odd length $q$ with $q?5$. In this paper, we further determine $\mathrm{ex}(n,H_{s,t})$ and the extremal graphs, where $s?0$ and $t?1$. Let $?(n,H)$ be the smallest integer such that, for all graphs $G$ on $n$ vertices, the edge set $E\left(G\right)$ can be partitioned into at most $?(n,H)$ parts, of which every part either is a single edge or forms a graph isomorphic to $H$. Pikhurko and Sousa conjectured that $?(n,H)=\mathrm{ex}(n,H)$ for $\chi \left(H\right)?3$ and all sufficiently large $n$. Liu and Sousa (2015) verified the conjecture for $H_{s,0}$. In this paper, we further verify Pikhurko and Sousa’s conjecture for $H_{s,t}$ with $s?0$ and $t?1$.     

On hypergraphs without loose cycles

Recently, Mubayi and Wang showed that for $r\ge 4$ and $?\ge 3$, the number of $n$-vertex $r$-graphs that do not contain any loose cycle of length $?$ is at most $2^{O\left(n^{r?1}{(logn)}^{(r?3)∕(r?2)}\right)}$. We improve this bound to $2^{O(n^{r?1}loglogn)}$.     

On 3-stable number conditions in n-connected claw-free graphs

For a subgraph $X$ of $G$, let ${\alpha }_G^3\left(X\right)$ be the maximum number of vertices of $X$ that are pairwise distance at least three in $G$. In this paper, we prove three theorems. Let $n$ be a positive integer, and let $H$ be a subgraph of an $n$-connected claw-free graph $G$. We prove that if $n\ge 2$, then either $H$ can be covered by a cycle in $G$, or there exists a cycle $C$ in $G$ such that ${\alpha }_G^3(H?V\left(C\right))\le {\alpha }_G^3\left(H\right)?n$. This result generalizes the result of Broersma and Lu that $G$ has a cycle covering all the vertices of $H$ if ${\alpha }_G^3\left(H\right)\le n$. We also prove that if $n\ge 1$, then either $H$ can be covered by a path in $G$, or there exists a path $P$ in $G$ such that ${\alpha }_G^3(H?V\left(P\right))\le {\alpha }_G^3\left(H\right)?n?1$. By using the second result, we prove the third result. For a tree $T$, a vertex of $T$ with degree one is called a leaf of $T$. For an integer $k\ge 2$, a tree which has at most $k$ leaves is called a $k$-ended tree. We prove that if ${\alpha }_G^3\left(H\right)\le n+k?1$, then $G$ has a $k$-ended tree covering all the vertices of $H$. This result gives a positive answer to the conjecture proposed by Kano et al. (2012).