Rainbow connection in 33-connected graphs

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this paper, we proved that $rc(G)\leq 3(n+1)/5$ for all $3$-connected graphs.Comment: 7 page

A Tur\'an-type problem on degree sequence

Given $p\geq 0$ and a graph $G$ whose degree sequence is $d_1,d_2,\ldots,d_n$, let $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Tur\'an-type problem for $e_p(G)$: given $p\geq 0$, how large can $e_p(G)$ be if $G$ has no subgraph of a particular type. Denote by $ex_p(n,H)$ the maximum value of $e_p(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $ex_1(n,H)=2ex(n,H)$, where $ex(n,H)$ denotes the classical Tur\'an number, i.e., the maximum number of edges among all $H$-free graphs with $n$ vertices. Pikhurko and Taraz generalize this Tur\'an-type problem: let $f$ be a non-negative increasing real function and $e_f(G)=\sum_{i=1}^n f(d_i)$, and then define $ex_f(n,H)$ as the maximum value of $e_f(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Observe that $ex_f(n,H)=ex(n,H)$ if $f(x)=x/2$, $ex_f(n,H)=ex_p(n,H)$ if $f(x)=x^p$. Bollob\'as and Nikiforov mentioned that it is important to study concrete functions. They gave an example $f(x)=\phi(k)={x\choose k}$, since $\sum_{i=1}^n{d_i\choose k}$ counts the $(k+1)$-vertex subgraphs of $G$ with a dominating vertex. Denote by $T_r(n)$ the $r$-partite Tur\'an graph of order $n$. In this paper, using the Bollob\'as--Nikiforov's methods, we give some results on $ex_{\phi}(n,K_{r+1})$ $(r\geq 2)$ as follows: for $k=1,2$, $ex_\phi(n,K_{r+1})=e_\phi(T_r(n))$; for each $k$, there exists a constant $c=c(k)$ such that for every $r\geq c(k)$ and sufficiently large $n$, $ex_\phi(n,K_{r+1})=e_\phi(T_r(n))$; for a fixed $(r+1)$-chromatic graph $H$ and every $k$, when $n$ is sufficiently large, we have $ex_\phi(n,H)=e_\phi(n,K_{r+1})+o(n^{k+1})$.Comment: 9 page

The strong rainbow vertex-connection of graphs

A vertex-colored graph $G$ is said to be rainbow vertex-connected if every two vertices of $G$ are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. If for every pair $u, v$ of distinct vertices, $G$ contains a rainbow $u-v$ geodesic, then $G$ is strong rainbow vertex-connected. The minimum number $k$ for which there exists a $k$-vertex-coloring of $G$ that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of $G$, denoted by $srvc(G)$. Observe that $rvc(G)\leq srvc(G)$ for any nontrivial connected graph $G$. In this paper, sharp upper and lower bounds of $srvc(G)$ are given for a connected graph $G$ of order $n$, that is, $0\leq srvc(G)\leq n-2$. Graphs of order $n$ such that $srvc(G)= 1, 2, n-2$ are characterized, respectively. It is also shown that, for each pair $a, b$ of integers with $a\geq 5$ and $b\geq (7a-8)/5$, there exists a connected graph $G$ such that $rvc(G)=a$ and $srvc(G)=b$.Comment: 10 page

Complete solution to a problem on the maximal energy of unicyclic bipartite graphs

The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Denote by $C_n$ the cycle, and $P_n^{6}$ the unicyclic graph obtained by connecting a vertex of $C_6$ with a leaf of $P_{n-6}$\,. Caporossi et al. conjecture that the unicyclic graph with maximal energy is $P_n^6$ for $n=8,12,14$ and $n\geq 16$. In``Y. Hou, I. Gutman and C. Woo, Unicyclic graphs with maximal energy, {\it Linear Algebra Appl.} {\bf 356}(2002), 27--36", the authors proved that $E(P_n^6)$ is maximal within the class of the unicyclic bipartite $n$-vertex graphs differing from $C_n$\,. And they also claimed that the energy of $C_n$ and $P_n^6$ is quasi-order incomparable and left this as an open problem. In this paper, by utilizing the Coulson integral formula and some knowledge of real analysis, especially by employing certain combinatorial techniques, we show that the energy of $P_n^6$ is greater than that of $C_n$ for $n=8,12,14$ and $n\geq 16$, which completely solves this open problem and partially solves the above conjecture.Comment: 8 page