A note on chromatic number of direct product of graphs
A note on cyclic chromatic number
A cyclic colouring of a graph G embedded in a surface is a vertex colouring of G in which any two distinct vertices sharing a face receive distinct colours. The cyclic chromatic number of G is the smallest number of colours in a cyclic colouring of G. Plummer and Toft in 1987 conjectured that for any 3-connected plane graph G with maximum face degree Δ*. It is known that the conjecture holds true for Δ* ≤ 4 and Δ* ≥ 18. The validity of the conjecture is proved in the paper for some special classes...
A note on edge-colourings avoiding rainbow and monochromatic .
A note on face coloring entire weightings of plane graphs
Given a weighting of all elements of a 2-connected plane graph G = (V,E, F), let f(α) denote the sum of the weights of the edges and vertices incident with the face _ and also the weight of _. Such an entire weighting is a proper face colouring provided that f(α) ≠ f(β) for every two faces α and _ sharing an edge. We show that for every 2-connected plane graph there is a proper face-colouring entire weighting with weights 1 through 4. For some families we improved 4 to 3
A note on graph coloring
A note on graph coloring extensions and list-colorings.
A note on graph colouring
A note on joins of additive hereditary graph properties
Let denote a set of additive hereditary graph properties. It is a known fact that a partially ordered set is a complete distributive lattice. We present results when a join of two additive hereditary graph properties in has a finite or infinite family of minimal forbidden subgraphs.
A note on -free precolouring with colours.
A note on line colorings of cubic graphs
A note on maximal -degenerate graphs
A note on Möbius inversion over power set lattices
In this paper, we establish a theorem on Möbius inversion over power set lattices which strongly generalizes an early result of Whitney on graph colouring.
A Note on Neighbor Expanded Sum Distinguishing Index
A total k-coloring of a graph G is a coloring of vertices and edges of G using colors of the set [k] = {1, . . . , k}. These colors can be used to distinguish the vertices of G. There are many possibilities of such a distinction. In this paper, we consider the sum of colors on incident edges and adjacent vertices.
A note on neighbour-distinguishing regular graphs total-weighting.
A note on on-line ranking number of graphs
A -ranking of a graph is a mapping such that each path with endvertices of the same colour contains an internal vertex with colour greater than . The ranking number of a graph is the smallest positive integer admitting a -ranking of . In the on-line version of the problem, the vertices of arrive one by one in an arbitrary order, and only the edges of the induced graph are known when the colour for the vertex has to be chosen. The on-line ranking number of a graph is the smallest...
A note on packing chromatic number of the square lattice.
A note on -colorings of graphs.
A note on radio antipodal colourings of paths
The radio antipodal number of a graph is the smallest integer such that there exists an assignment satisfying for every two distinct vertices and of , where is the diameter of . In this note we determine the exact value of the antipodal number of the path, thus answering the conjecture given in [G. Chartrand, D. Erwin and P. Zhang, Math. Bohem. 127 (2002), 57–69]. We also show the connections between this colouring and radio labelings.
A note on stable sets and colorings of graphs
A note on the chromatic number of the alternative negation of two graphs