The choice number of a graph G is the smallest integer k such that for every assignment of a list L(v) of k colors to each vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from L(v). We present upper and lower bounds on the choice number of complete multipartite graphs with partite classes of equal sizes and complete r-partite graphs with r-1 partite classes of order two.

We derive an upper bound on the number of vertices in graphs of diameter 3 and given degree arising from Abelian lifts of dipoles with loops and multiple edges.

Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph $G$, a hereditary graph property $\mathcal{P}$ and $l\ge 1$ we define ${\chi}_{\mathcal{P},l}^{\text{'}}\left(G\right)$ to be the minimum number of colours needed to properly colour the edges of $G$, such that any subgraph of $G$ induced by edges coloured by (at most) $l$ colours is in $\mathcal{P}$. We present a necessary and sufficient condition for the existence of ${\chi}_{\mathcal{P},l}^{\text{'}}\left(G\right)$. We focus on edge-colourings of graphs with respect to the hereditary properties...

For positive integers Δ and D we define nΔ,D to be the largest number of vertices in an outerplanar graph of given maximum degree Δ and diameter D. We prove that [...] nΔ,D=ΔD2+O (ΔD2−1) ${n}_{\Delta ,D}={\Delta}^{\frac{D}{2}}+O\left({\Delta}^{\frac{D}{2}-1}\right)$ is even, and [...] nΔ,D=3ΔD−12+O (ΔD−12−1) ${n}_{\Delta ,D}=3{\Delta}^{\frac{D-1}{2}}+O\left({\Delta}^{\frac{D-1}{2}-1}\right)$ if D is odd. We then extend our result to maximal outerplanar graphs by showing that the maximum number of vertices in a maximal outerplanar graph of maximum degree Δ and diameter D asymptotically equals nΔ,D.

For graphs $G$, ${F}_{1}$, ${F}_{2}$, we write $G\to ({F}_{1},{F}_{2})$ if for every red-blue colouring of the edge set of $G$ we have a red copy of ${F}_{1}$ or a blue copy of ${F}_{2}$ in $G$. The size Ramsey number $\widehat{r}({F}_{1},{F}_{2})$ is the minimum number of edges of a graph $G$ such that $G\to ({F}_{1},{F}_{2})$. Erdős and Faudree proved that for the cycle ${C}_{n}$ of length $n$ and for $t\ge 2$ matchings $t{K}_{2}$, the size Ramsey number $\widehat{r}(t{K}_{2},{C}_{n})<n+(4t+3)\sqrt{n}$. We improve their upper bound for $t=2$ and $t=3$ by showing that $\widehat{r}(2{K}_{2},{C}_{n})\le n+2\sqrt{3n}+9$ for $n\ge 12$ and $\widehat{r}(3{K}_{2},{C}_{n})<n+6\sqrt{n}+9$ for $n\ge 25$.

A directed Cayley graph $C(\Gamma ,X)$ is specified by a group $\Gamma $ and an identity-free generating set $X$ for this group. Vertices of $C(\Gamma ,X)$ are elements of $\Gamma $ and there is a directed edge from the vertex $u$ to the vertex $v$ in $C(\Gamma ,X)$ if and only if there is a generator $x\in X$ such that $ux=v$. We study graphs $C(\Gamma ,X)$ for the direct product ${Z}_{m}\times {Z}_{n}$ of two cyclic groups ${Z}_{m}$ and ${Z}_{n}$, and the generating set $X=\left\{\right(0,1),(1,0),(2,0),\cdots ,(p,0\left)\right\}$. We present resolving sets which yield upper bounds on the metric dimension of these graphs for $p=2$ and $3$.

For graphs F, G and H, we write F → (G,H) to mean that any red-blue coloring of the edges of F contains a red copy of G or a blue copy of H. The graph F is Ramsey (G,H)-minimal if F → (G,H) but F* ↛ (G,H) for any proper subgraph F* ⊂ F. We present an infinite family of Ramsey $({K}_{1,2},C\u2084)$-minimal graphs of any diameter ≥ 4.

The Gutman index and the edge-Wiener index have been extensively investigated particularly in the last decade. An important stream of re- search on graph indices is to bound indices in terms of the order and other parameters of given graph. In this paper we present asymptotically sharp upper bounds on the Gutman index and the edge-Wiener index for graphs of given order and vertex-connectivity κ, where κ is a constant. Our results substantially generalize and extend known results in the area.

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