Characterizing All Trees with Locating-chromatic Number 3

Let $c$ be a proper $k$-coloring of a connected graph $G$. Let $\Pi = \{S_{1}, S_{2},\ldots, S_{k}\}$ be the induced partition of $V(G)$ by $c$, where $S_{i}$ is the partition class having all vertices with color $i$.The color code $c_{\Pi}(v)$ of vertex $v$ is the ordered$k$-tuple $(d(v,S_{1}), d(v,S_{2}),\ldots, d(v,S_{k}))$, where$d(v,S_{i})= \hbox{min}\{d(v,x)|x \in S_{i}\}$, for $1\leq i\leq k$.If all vertices of $G$ have distinct color codes, then $c$ iscalled a locating-coloring of $G$.The locating-chromatic number of $G$, denoted by $\chi_{L}(G)$, isthe smallest $k$ such that $G$ posses a locating $k$-coloring. Clearly, any graph of order $n \geq 2$ have locating-chromatic number $k$, where $2 \leq k \leq n$. Characterizing all graphswith a certain locating-chromatic number is a difficult problem. Up to now, we have known allgraphs of order $n$ with locating chromatic number $2, n-1,$ or $n$.In this paper, we characterize all trees whose locating-chromatic number $3$. We also give a family of trees with locating-chromatic number 4

Restricted Size Ramsey Number for Path of Order Three Versus Graph of Order Five

Let $G$ and $H$ be simple graphs. The Ramsey number for a pair of graph $G$ and $H$ is the smallest number $r$ such that any red-blue coloring of edges of $K_r$ contains a red subgraph $G$ or a blue subgraph $H$. The size Ramsey number for a pair of graph $G$ and $H$ is the smallest number $\hat{r}$ such that there exists a graph $F$ with size $\hat{r}$ satisfying the property that any red-blue coloring of edges of $F$ contains a red subgraph $G$ or a blue subgraph $H$. Additionally, if the order of $F$ in the size Ramsey number is $r(G,H)$, then it is called the restricted size Ramsey number. In 1983, Harary and Miller started to find the (restricted) size Ramsey number for any pair of small graphs with order at most four. Faudree and Sheehan (1983) continued Harary and Miller\u27s works and summarized the complete results on the (restricted) size Ramsey number for any pair of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for any pair of small forests with order at most five. To continue their works, we investigate the restricted size Ramsey number for a path of order three versus connected graph of order five

On Energy, Laplacian Energy and PP-fold Graphs

For a graph $G$ having adjacency spectrum ($A$-spectrum) $\lambda_n\leq\lambda_{n-1}\leq\cdots\leq\lambda_1$ and Laplacian spectrum ($L$-spectrum) $0=\mu_n\leq\mu_{n-1}\leq\cdots\leq\mu_1$, the energy is defined as $E(G)=\sum_{i=1}^{n}|\lambda_i|$ and the Laplacian energy is defined as $LE(G)=\sum_{i=1}^{n}|\mu_i-\frac{2m}{n}|$. In this paper, we give upper and lower bounds for the energy of $KK_n^j,~1\leq j \leq n$ and as a consequence we generalize a result of Stevanovic et al. [More on the relation between Energy and Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. {\bf 61} (2009) 395-401]. We also consider strong double graph and strong $p$-fold graph to construct some new families of graphs $G$ for which E(G)> LE(G)

On Size Multipartite Ramsey Numbers for Stars Versus Paths and Cycles

Let $K_{l\times t}$ be a complete, balanced, multipartite graph consisting of $l$ partite sets and $t$ vertices in each partite set. For given two graphs $G_1$ and $G_2$, and integer $j\geq 2$, the size multipartite Ramsey number $m_j(G_1,G_2)$ is the smallest integer $t$ such that every factorization of the graph $K_{j\times t}:=F_1\oplus F_2$ satisfies the following condition: either $F_1$ contains $G_1$ or $F_2$ contains $G_2$. In 2007, Syafrizal et al. determined the size multipartite Ramsey numbers of paths $P_n$ versus stars, for $n=2,3$ only. Furthermore, Surahmat et al. (2014) gave the size tripartite Ramsey numbers of paths $P_n$ versus stars, for $n=3,4,5,6$. In this paper, we investigate the size tripartite Ramsey numbers of paths $P_n$ versus stars, with all $n\geq 2$. Our results complete the previous results given by Syafrizal et al. and Surahmat et al. We also determine the size bipartite Ramsey numbers $m_2(K_{1,m},C_n)$ of stars versus cycles, for $n\geq 3,m\geq 2$

Bilangan Ramsey Sisi dari R(p3,pn)

Pada paper ini akan ditunjukkan bahwa bilangan Ramsey sisi dari r(P3,Pn) untuk n = 13, 14, 15 adalah 20, 23, 24. Ditunjukkan pula bahwa r(P3,pn)=r(P3,Pk)+r(P3,P1) dengan n = k+l-1 untuk n ganjil dan k, l genap. Kata kunci: Bilangan Ramsey sisi, Graph lintasa