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تسجيل مستخدم جديدLovasz extension and graph cut
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A set-pair Lovasz extension is established to construct equivalent continuous optimization problems for graph $k$-cut problems.
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In 1972, Erd{o}s - Faber - Lov{a}sz (EFL) conjectured that, if $textbf{H}$ is a linear hypergraph consisting of $n$ edges of cardinality $n$, then it is possible to color the vertices with $n$ colors so that no two vertices with the same color are in
the same edge. In 1978, Deza, Erd{o}s and Frankl had given an equivalent version of the same for graphs: Let $G= bigcup_{i=1}^{n} A_i$ denote a graph with $n$ complete graphs $A_1, A_2,$ $ dots , A_n$, each having exactly $n$ vertices and have the property that every pair of complete graphs has at most one common vertex, then the chromatic number of $G$ is $n$. The clique degree $d^K(v)$ of a vertex $v$ in $G$ is given by $d^K(v) = |{A_i: v in V(A_i), 1 leq i leq n}|$. In this paper we give a method for assigning colors to the graphs satisfying the hypothesis of the Erdos - Faber - Lovasz conjecture using intersection matrix of the cliques $A_i$s of $G$ and clique degrees of the vertices of $G$. Also, we give theoretical proof of the conjecture for some class of graphs. In particular we show that: 1. If $G$ is a graph satisfying the hypothesis of the Conjecture 1.2 and every $A_i$ ($1 leq i leq n$) has at most $sqrt{n}$ vertices of clique degree greater than 1, then $G$ is $n$-colorable. 2. If $G$ is a graph satisfying the hypothesis of the Conjecture 1.2 and every $A_i$ ($1 leq i leq n$) has at most $left lceil {frac{n+d-1}{d}} right rceil$ vertices of clique degree greater than or equal to $d$ ($2leq d leq n$), then $G$ is $n$-colorable.
The ErdH{o}s-Faber-Lov{a}sz conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. In this paper, we prove this conjecture for every large $n$. We also provide stabili
We consider two-dimensional Schroedinger operators with an attractive potential in the form of a channel of a fixed profile built along an unbounded curve composed of a circular arc and two straight semi-lines. Using a test-function argument with hel
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We briefly review the diffraction of quasicrystals and then give an elementary alternative proof of the diffraction formula for regular cut-and-project sets, which is based on Bochners theorem from Fourier analysis. This clarifies a common view that
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A hole in a graph is an induced cycle of length at least $4$. Let $sge2$ and $tge2$ be integers. A graph $G$ is $(s,t)$-splittable if $V(G)$ can be partitioned into two sets $S$ and $T$ such that $chi(G[S ]) ge s$ and $chi(G[T ]) ge t$. The well-know
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