اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA note on Edge Isoperimetric Numbers and Regular Graphs
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This note resolves an open problem asked by Bezrukov in the open problem session of IWOCA 2014. It shows an equivalence between regular graphs and graphs for which a sequence of invariants presents some symmetric property. We extend this result to a few other sequences.
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In this short note, we show two NP-completeness results regarding the emph{simultaneous representation problem}, introduced by Lubiw and Jampani. The simultaneous representation problem for a given class of intersection graphs asks if some $k$ graphs
can be represented so that every vertex is represented by the same interval in each representation. We prove that it is NP-complete to decide this for the class of interval and circular-arc graphs in the case when $k$ is a part of the input and graphs are not in a sunflower position.
For a graph $G$ and integer $qgeq 2$, an edge $q$-coloring of $G$ is an assignment of colors to edges of $G$, such that edges incident on a vertex span at most $q$ distinct colors. The maximum edge $q$-coloring problem seeks to maximize the number of
colors in an edge $q$-coloring of a graph $G$. The problem has been studied in combinatorics in the context of {em anti-Ramsey} numbers. Algorithmically, the problem is NP-Hard for $qgeq 2$ and assuming the unique games conjecture, it cannot be approximated in polynomial time to a factor less than $1+1/q$. The case $q=2$, is particularly relevant in practice, and has been well studied from the view point of approximation algorithms. A $2$-factor algorithm is known for general graphs, and recently a $5/3$-factor approximation bound was shown for graphs with perfect matching. The algorithm (which we refer to as the matching based algorithm) is as follows: Find a maximum matching $M$ of $G$. Give distinct colors to the edges of $M$. Let $C_1,C_2,ldots, C_t$ be the connected components that results when M is removed from G. To all edges of $C_i$ give the $(|M|+i)$th color. In this paper, we first show that the approximation guarantee of the matching based algorithm is $(1 + frac {2} {delta})$ for graphs with perfect matching and minimum degree $delta$. For $delta ge 4$, this is better than the $frac {5} {3}$ approximation guarantee proved in {AAAP}. For triangle free graphs with perfect matching, we prove that the approximation factor is $(1 + frac {1}{delta - 1})$, which is better than $5/3$ for $delta ge 3$.
A graph is IC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share no common end vertex. IC-planarity specializes both NIC-planarity, which allows a pair of crossing edges to sh
are at most one vertex, and 1-planarity, where each edge may be crossed at most once. We show that there are infinitely maximal IC-planar graphs with n vertices and 3n-5 edges and thereby prove a tight lower bound on the density of this class of graphs.
An edge-coloring of a graph $G$ with colors $1,2,ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if i
t has an interval $t$-coloring for some positive integer $t$. For an interval colorable graph $G$, $W(G)$ denotes the greatest value of $t$ for which $G$ has an interval $t$-coloring. It is known that the complete graph is interval colorable if and only if the number of its vertices is even. However, the exact value of $W(K_{2n})$ is known only for $n leq 4$. The second author showed that if $n = p2^q$, where $p$ is odd and $q$ is nonnegative, then $W(K_{2n}) geq 4n-2-p-q$. Later, he conjectured that if $n in mathbb{N}$, then $W(K_{2n}) = 4n - 2 - leftlfloorlog_2{n}rightrfloor - left | n_2 right |$, where $left | n_2 right |$ is the number of $1$s in the binary representation of $n$. In this paper we introduce a new technique to construct interval colorings of complete graphs based on their 1-factorizations, which is used to disprove the conjecture, improve lower and upper bounds on $W(K_{2n})$ and determine its exact values for $n leq 12$.
Clique-width is a complexity measure of directed as well as undirected graphs. Rank-width is an equivalent complexity measure for undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertex-minor re
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