534 research outputs found
Largest Digraphs Contained IN All N-tournaments
Let f(n) (resp. g(n)) be the largest m such that there is a digraph (resp. a spanning weakly connected digraph) on n-vertices and m edges which is a subgraph of every tournament on n-vertices. We prove that n log2 n--cxn>=f(n) ~_g(n) ~- n log ~ n--c..n loglog n
Switching Reconstruction of Digraphs
Switching about a vertex in a digraph means to reverse the direction of every
edge incident with that vertex. Bondy and Mercier introduced the problem of
whether a digraph can be reconstructed up to isomorphism from the multiset of
isomorphism types of digraphs obtained by switching about each vertex. Since
the largest known non-reconstructible oriented graphs have 8 vertices, it is
natural to ask whether there are any larger non-reconstructible graphs. In this
paper we continue the investigation of this question. We find that there are
exactly 44 non-reconstructible oriented graphs whose underlying undirected
graphs have maximum degree at most 2. We also determine the full set of
switching-stable oriented graphs, which are those graphs for which all
switchings return a digraph isomorphic to the original
Hitting minors, subdivisions, and immersions in tournaments
The Erd\H{o}s-P\'osa property relates parameters of covering and packing of
combinatorial structures and has been mostly studied in the setting of
undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim,
and Seymour to show that, for every directed graph (resp.
strongly-connected directed graph ), the class of directed graphs that
contain as a strong minor (resp. butterfly minor, topological minor) has
the vertex-Erd\H{o}s-P\'osa property in the class of tournaments. We also prove
that if is a strongly-connected directed graph, the class of directed
graphs containing as an immersion has the edge-Erd\H{o}s-P\'osa property in
the class of tournaments.Comment: Accepted to Discrete Mathematics & Theoretical Computer Science.
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