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The odd-girth of a graph is the length of a shortest odd circuit. A conjecture by Pavol Hell about circular coloring is solved in this article by showing that there is a function f ( ) for each : 0 < < 1 such that, if the odd-girth of a planar graph G is at least f ( ), then G is (2 + )-colorable. Note that the function f ( ) is independent of the graph G and → 0 if and only if f ( ) → ∞. A key lemma, called the folding lemma, is proved that provides a reduction method, which maintains the odd-girth of planar graphs. This lemma is expected to have applications in related problems.

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We examine subgraphs of oriented graphs in the context of oriented coloring that are analogous to cliques in traditional vertex coloring. Bounds on the sizes of these subgraphs are given for planar, outerplanar, and series-parallel graphs. In particular, the main result of the paper is that a planar graph cannot contain an induced subgraph D with more than 36 vertices such that each pair of vertices in D are joined by a directed path of length at most two.

A secure (total ) dominating set of a graph G = (V, E) is a (total) dominating set X ⊆ V with the property that for each u ∈ V − X, there exists x ∈ X adjacent to u such that (X − {x}) ∪ {u} is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total ) domination number γ s (G) (γ st (G)). We characterize graphs with equal total and secure total domination numbers. We show that if G has minimum degree at least two, then γ st (G) ≤ γ s (G). We also show that γ st (G) is at most twice the clique covering number of G, and less than three times the independence number. With the exception of the independence number bound, these bounds are sharp.

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