A note on arbitrarily vertex decomposable graphs

A graph $G$ of order $n$ is said to be arbitrarily vertex decomposable if for each sequence $(n_{1},\ldots,n_k)$ of positive integers such that $n_{1}+\ldots+n_{k}=n$ there exists a partition $(V_{1},\ldots,V_{k})$ of the vertex set of $G$ such that for each $i \in \{1,\ldots,k\}$, $V_{i}$ induces a connected subgraph of $G$ on $n_i$ vertices. In this paper we show that if $G$ is a two-connected graph on $n$ vertices with the independence number at most $\lceil n/2\rceil$ and such that the degree sum of any pair of non-adjacent vertices is at least $n-3$, then $G$ is arbitrarily vertex decomposable. We present another result for connected graphs satisfying a similar condition, where the bound $n-3$ is replaced by $n-2$

A note on Arbitrarily vertex decomposable graphs, Opuscula Mathematica 26

A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n1,...,nk) of positive integers such that n1 +... + nk = n there exists a partition (V1,...,Vk) of the vertex set of G such that for each i ∈ {1,...,k}, Vi induces a connected subgraph of G on ni vertices. In this paper we show that if G is a two-connected graph on n vertices with the independence number at most ⌈n/2 ⌉ and such that the degree sum of any pair of nonadjacent vertices is at least n − 3, then G is arbitrarily vertex decomposable. We present another result for connected graphs satisfying a similar condition where the bound n − 3 is replaced by n − 2.

On a total version of 1,2,3 Conjecture

A total k-coloring of a graph G is a coloring of vertices and edges of G using colors of the set {1,. .. , k}. These colors can be used to distinguish adjacent vertices of G. There are many possibilities of such a distinction. In this paper we focus on the one by the full sum of colors of a vertex, i.e. the sum of the color of the vertex, the colors on its incident edges and the colors on its adjacent vertices. This way of distinguishing vertices has similar properties to the method when we only use incident edge colors and to the corresponding 1-2-3 Conjecture

Note on group irregularity strength of disconnected graphs

We investigate the group irregularity strength (sg(G)) of graphs, i.e. the smallest value of s such that taking any Abelian group of order s, there exists a function f : E(G) → such that the sums of edge labels at every vertex are distinct. So far it was not known if sg(G) is finite for disconnected graphs. In the paper we present some upper bound for all graphs. Moreover we give the exact values and bounds on sg(G) for disconnected graphs without a star as a component

A note on a new condition implying pancyclism

We first show that if a 2-connected graph G of order n is such that for each two vertices u and v such that δ = d(u) and d(v) < n/2 the edge uv belongs to E(G), then G is hamiltonian. Next, by using this result, we prove that a graph G satysfying the above condition is either pancyclic or isomorphic to $K_{n/2,n/2}$