A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v’s neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic num...

Refining a bound by Lih, Wang and Zhu, we prove that if the square G2 of a K4-minor-free graph Gwith maximum degree∆ > 6 does not contain a complete subgraph on ⌊ 3 2∆ ⌋ + 1 vertices, then G2 is ⌊ 3 2∆ ⌋ -colorable. © 2009 Elsevier B.V. All rights reserved.

The preparation of activated carbons (ACs) from the thermal decomposition of mixtures of orthophosphoric acid (PA) and either as-received softwood Kraft lignin, KL, or demineralised one, KL(d), has been investigated. Activation with PA has been studied for a PA/lignin ratio of 1 (dry ash-free basis) and 1h carbonisation time at final temperatures of 400, 500 and 600 degrees C. The yield, surfac...

A well-quasi-order is an order which contains no infinite decreasing sequence and no infinite collection of incomparable elements. In this paper, we consider graph classes defined by excluding one graph as contraction. More precisely, we give a complete characterization of graphs H such that the class of H-contraction-free graphs is well-quasi-ordered by the contraction relation. This result is...

In this article we present a structural characterization of graphs without K5 and the octahedron as a minor. We introduce semiplanar graphs as arbitrary sums of planar graphs, and give their characterization in terms of excluded minors. Some other excluded minor theorems for 3-connected minors are shown.

We prove that every graph of minimum degree at least r and girth at least 186 contains a subdivision of Krþ1 and that for r5435 a girth of at least 15 suffices. This implies that the conjecture of Haj ! os that every graph of chromatic number at least r contains a subdivision of Kr (which is false in general) is true for graphs of girth at least 186 (or 15 if r5436). More generally, we show tha...

A graph G is said to be 1-perfectly orientable if it has an orientation D such that for every vertex v ∈ V (G), the out-neighborhood of v in D is a clique in G. We characterize the class of 1-perfectly orientable K4-minor-free graphs. As a consequence we obtain a characterization of 1-perfectly orientable outerplanar graphs.

Let G be a graph and p ∈ [1,∞]. The parameter fp(G) is the least integer k such that for all m and all vectors (rv)v∈V (G) ⊆ R , there exist vectors (qv)v∈V (G) ⊆ R k satisfying ‖rv − rw‖p = ‖qv − qw‖p, for all vw ∈ E(G). It is easy to check that fp(G) is always finite and that it is minor monotone. By the graph minor theorem of Robertson and Seymour [9], there are a finite number of excluded m...

A graph is an apex if it contains a vertex whose deletion leaves planar graph. The family of graphs minor-closed and so characterized by finite list minor-minimal non-members. long-standing problem determining this obstructions remains open. This paper determines the 133 minor-minimal, non-apex that have connectivity two.

It is proved that if G is a K2,3-minor-free graph with maximum degree ∆, then ∆ + 1 6 χ(G) 6 ch(G) 6 ∆ + 2 if ∆ > 3, and ch(G) = χ(G) = ∆ + 1 if ∆ > 6. All inequalities here are sharp, even for outerplanar graphs.