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When a line graph associated to annihilating-ideal graph of a lattice is planar or projective

Atossa Parsapour, Khadijeh Ahmad Javaheri (2018)

Czechoslovak Mathematical Journal

Let ( L , , ) be a finite lattice with a least element 0. 𝔸 G ( L ) is an annihilating-ideal graph of L in which the vertex set is the set of all nontrivial ideals of L , and two distinct vertices I and J are adjacent if and only if I J = 0 . We completely characterize all finite lattices L whose line graph associated to an annihilating-ideal graph, denoted by 𝔏 ( 𝔸 G ( L ) ) , is a planar or projective graph.

WORM Colorings of Planar Graphs

J. Czap, S. Jendrol’, J. Valiska (2017)

Discussiones Mathematicae Graph Theory

Given three planar graphs F,H, and G, an (F,H)-WORM coloring of G is a vertex coloring such that no subgraph isomorphic to F is rainbow and no subgraph isomorphic to H is monochromatic. If G has at least one (F,H)-WORM coloring, then W−F,H(G) denotes the minimum number of colors in an (F,H)-WORM coloring of G. We show that (a) W−F,H(G) ≤ 2 if |V (F)| ≥ 3 and H contains a cycle, (b) W−F,H(G) ≤ 3 if |V (F)| ≥ 4 and H is a forest with Δ (H) ≥ 3, (c) W−F,H(G) ≤ 4 if |V (F)| ≥ 5 and H is a forest with...

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