Julien Bensmail , Sandip Das , Soumen Nandi , Ayan Nandy , Théo Pierron , Swathy Prabhu , Sagnik Sen
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引用次数: 0
Abstract
A proper n-coloring of a graph G is an assignment of colors from to its vertices such that no two adjacent vertices get assigned the same color. The chromatic number of G, denoted by , refers to the smallest n such that G admits a proper n-coloring. This notion naturally extends to edge-colorings (resp. total-colorings) when edges (resp. both vertices and edges) are to be colored, and this provides other parameters of G: its chromatic index and its total chromatic number .
These coloring notions are among the most fundamental ones of the graph coloring theory. As such, they gave birth to hundreds of studies dedicated to several of their aspects, including generalizations to more general structures such as oriented graphs. They include notably the notions of oriented n-colorings and oriented n-arc-colorings, which stand as natural extensions of their undirected counterparts, and which have been receiving increasing attention.
Our goal is to introduce a missing piece in this line of work, namely the oriented counterparts of proper n-total-colorings and total chromatic number. We first define these notions and show that they share properties and connections with oriented (arc) colorings that are reminiscent of those shared by their undirected counterparts. We then focus on understanding the oriented total chromatic number of particular types of oriented graphs, such as oriented forests, cycles, and some planar graphs. Finally, we establish a full complexity dichotomy for the problem of determining whether an oriented graph is totally k-colorable.
Throughout this work, each of our results is compared to what is known regarding the oriented chromatic number and oriented chromatic index. We also disseminate some directions for further research on the oriented total chromatic number.
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.