A note on neighbour-distinguishing regular graphs total-weighting.
Irregularity strength of regular graphs.
Total weight choosability of graphs.
Vertex-distinguishing edge-colorings of linear forests
In the PhD thesis by Burris (Memphis (1993)), a conjecture was made concerning the number of colors c(G) required to edge-color a simple graph G so that no two distinct vertices are incident to the same multiset of colors. We find the exact value of c(G) - the irregular coloring number, and hence verify the conjecture when G is a vertex-disjoint union of paths. We also investigate the point-distinguishing chromatic index, χ₀(G), where sets, instead of multisets, are required to be distinct, and...
On An Extremal Problem In The Class Of Bipartite 1-Planar Graphs
A graph G = (V, E) is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite 1-planar graphs with prescribed numbers of vertices in partite sets. Bipartite 1-planar graphs are known to have at most 3n − 8 edges, where n denotes the order of a graph. We show that maximal-size bipartite 1-planar graphs which are almost balanced have not significantly fewer edges than indicated by this upper bound, while the same is not true...
Rainbow Connection In Sparse Graphs
An edge-coloured connected graph G = (V,E) is called rainbow-connected if each pair of distinct vertices of G is connected by a path whose edges have distinct colours. The rainbow connection number of G, denoted by rc(G), is the minimum number of colours such that G is rainbow-connected. In this paper we prove that rc(G) ≤ k if |V (G)| = n and for all integers n and k with n − 6 ≤ k ≤ n − 3. We also show that this bound is tight.
Arbitrarily vertex decomposable caterpillars with four or five leaves
A graph G of order n is called arbitrarily vertex decomposable if for each sequence (a₁,...,aₖ) of positive integers such that a₁+...+aₖ = n there exists a partition (V₁,...,Vₖ) of the vertex set of G such that for each i ∈ 1,...,k, induces a connected subgraph of G on vertices. D. Barth and H. Fournier showed that if a tree T is arbitrarily vertex decomposable, then T has maximum degree at most 4. In this paper we give a complete characterization of arbitrarily vertex decomposable caterpillars...
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