On k-geodetic digraphs with excess one

<pre>A <span>k</span>-geodetic digraph <span>G</span> is a digraph in which, for every pair of vertices <span>u</span> and <span>v</span> (not necessarily distinct), there is at most one walk of length at most <span>k</span> from <span>u</span> to <span>v</span>. If the diameter of <span>G</span> is <span>k</span>, we say that <span>G</span> is strongly geodetic. Let <span>N(d,k)</span> be the smallest possible order for a <span>k</span>-geodetic digraph of minimum out-degree <span>d</span>, then <span>N(d,k) is at most 1+d+d^2+...+d^k=M(d,k)</span>, where <span>M(d,k)</span> is the Moore bound obtained if and only if <span>G</span> is strongly geodetic. Thus strongly geodetic digraphs only exist for <span>d=1</span> or <span>k=1</span>, hence for <span>d,k &gt;1</span> we wish to determine if <span>N(d,k)=M(d,k)+1</span> is possible. A <span>k</span>-geodetic digraph with minimum out-degree <span>d</span> and order <span>M(d,k)+1</span> is denoted as a <span>(d,k,1)</span>-digraph or said to have excess <span>1</span>.</pre><pre>In this paper we will prove that a <span>(d,k,1)</span>-digraph is always out-regular and that if it is not in-regular, then it must have <span>2</span> vertices of in-degree less than <span>d</span>, <span>d</span> vertices of in-degree <span>d+1</span> and the remaining vertices will have in-degree <span>d</span>.</pre><pre>Furthermore we will prove there exist no <span>(2,2,1)</span>-digraphs and no diregular <span>(2,k,1)</span>-digraphs for <span>k&gt; 2</span>.</pre

On k-geodetic digraphs with excess one

<pre>A <span>k</span>-geodetic digraph <span>G</span> is a digraph in which, for every pair of vertices <span>u</span> and <span>v</span> (not necessarily distinct), there is at most one walk of length at most <span>k</span> from <span>u</span> to <span>v</span>. If the diameter of <span>G</span> is <span>k</span>, we say that <span>G</span> is strongly geodetic. Let <span>N(d,k)</span> be the smallest possible order for a <span>k</span>-geodetic digraph of minimum out-degree <span>d</span>, then <span>N(d,k) is at most 1+d+d^2+...+d^k=M(d,k)</span>, where <span>M(d,k)</span> is the Moore bound obtained if and only if <span>G</span> is strongly geodetic. Thus strongly geodetic digraphs only exist for <span>d=1</span> or <span>k=1</span>, hence for <span>d,k &gt;1</span> we wish to determine if <span>N(d,k)=M(d,k)+1</span> is possible. A <span>k</span>-geodetic digraph with minimum out-degree <span>d</span> and order <span>M(d,k)+1</span> is denoted as a <span>(d,k,1)</span>-digraph or said to have excess <span>1</span>.</pre><pre>In this paper we will prove that a <span>(d,k,1)</span>-digraph is always out-regular and that if it is not in-regular, then it must have <span>2</span> vertices of in-degree less than <span>d</span>, <span>d</span> vertices of in-degree <span>d+1</span> and the remaining vertices will have in-degree <span>d</span>.</pre><pre>Furthermore we will prove there exist no <span>(2,2,1)</span>-digraphs and no diregular <span>(2,k,1)</span>-digraphs for <span>k&gt; 2</span>.</pre

Subdigraphs of almost Moore digraphs induced by fixpoints of an automorphism

<pre>The degree/diameter problem for directed graphs is the problem of determining the largest possible order for a digraph with given maximum out-degree <span>d</span> and diameter <span>k</span>. An upper bound is given by the Moore bound <span>M(d,k)=1+d+d^2+...+d^k$</span> and almost Moore digraphs are digraphs with maximum out-degree <span>d</span>, diameter <span>k</span> and order <span>M(d,k)-1</span>.</pre><pre> </pre><pre>In this paper we will look at the structure of subdigraphs of almost Moore digraphs, which are induced by the vertices fixed by some automorphism <span>varphi</span>. If the automorphism fixes at least three vertices, we prove that the induced subdigraph is either an almost Moore digraph or a diregular <span>k</span>-geodetic digraph of degree <span>d'&lt;d-1</span>, order <span>M(d',k)+1</span> and diameter <span>k+1</span>.</pre><pre> </pre><pre>As it is known that almost Moore digraphs have an automorphism <span>r</span>, these results can help us determine structural properties of almost Moore digraphs, such as how many vertices of each order there are with respect to <span>r</span>. We determine this for <span>d=4</span> and <span>d=5</span>, where we prove that except in some special cases, all vertices will have the same order.</pre

Totally antimagic total graphs

For a graph G a bijection from the vertex set and the edge set of G to the set {1, 2, ..., |V(G)| + |E(G)|} is called a total labeling of G. The edge-weight of an edge is the sum of the label of the edge and the labels of the end vertices of that edge. The vertex-weight of a vertex is the sum of the label of the vertex and the labels of all the edges incident with that vertex. A total labeling is called edge-antimagic total (vertexantimagic total) if all edge-weights (vertex-weights) are pairwise distinct. If a labeling is simultaneously edge-antimagic total and vertex-antimagic total it is called a totally antimagic total labeling. A graph that admits totally antimagic total labeling is called a totally antimagic total graph. In this paper we deal with the problem of finding totally antimagic total labeling of some classes of graphs. We prove that paths, cycles, stars, double-stars and wheels are totally antimagic total. We also show that a union of regular totally antimagic total graphs is a totally antimagic total graph