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List Improper Colourings of Planar Graphs
  • R. Škrekovski
  • Mathematics
    Combinatorics, Probability and Computing
  • 1 May 1999
A graph G is m-choosable with impropriety d, or simply (m, d)*-choosable, if for every list assignment L, where [mid ]L(v)[mid ][ges ]m for every v∈V(G), there exists an L-colouring of G such that
A Theorem about the Channel Assignment Problem
All balanced list channel assignment problems (G,L,w) which admit a proper coloring are characterized, which means that each graph with maximum degree $\Delta\ge 2$ has an L(2,1)-labeling using integers $0,\ldots,\Delta^2+\Delta-1$.
Star Edge Coloring of Some Classes of Graphs
Tight upper bounds for trees and subcubic outerplanar graphs are established and an upper bound for outerplanars graphs is derived.
The Grötzsch Theorem for the Hypergraph of Maximal Cliques
The Grotzsch Theorem is extended to list colorings by proving that the clique hypergraph of every planar graph is 3-colorable and 4-choosability of ${\cal H}(G)$ is established for the class of locally planar graphs on arbitrary surfaces.
List Total Colourings of Graphs
The concept of list total colourings is studied and it is proved that every multigraph of maximum degree 3 is 5-total-choosable and the total choice number of a graph is equal to its total chromatic number.
Cycles Intersecting Edge-Cuts of Prescribed Sizes
The concept of a coverable set of integers is introduced and a number of questions are discussed, some of which are related to classical problems of graph theory such as Tutte's $4$-flow conjecture and the Dominating Cycle Conjecture.
An improved bound on the largest induced forests for triangle-free planar graphs
It is proved that every planar triangle-free graph of order n has a subset of vertices that induces a forest of size at least (71n + 72)/128 and poses some questions regarding planar graphs of higher girth.
Mostar index
We propose and investigate a new bond-additive structural invariant as a measure of peripherality in graphs. We first determine its extremal values and characterize extremal trees and unicyclic