### Nowhere-zero $k$-flows of supergraphs.

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In this paper Gallai’s inequality on the number of edges in critical graphs is generalized for reducible additive induced-hereditary properties of graphs in the following way. Let $\u2081,\u2082,...,\u2096$ (k ≥ 2) be additive induced-hereditary properties, $=\u2081\circ \u2082\circ ...\circ \u2096$ and $\delta ={\sum}_{i=1}^{k}\delta {(}_{i})$. Suppose that G is an -critical graph with n vertices and m edges. Then 2m ≥ δn + (δ-2)/(δ²+2δ-2)*n + (2δ)/(δ²+2δ-2) unless = ² or $G={K}_{\delta +1}$. The generalization of Gallai’s inequality for -choice critical graphs is also presented.

We show for every k ≥ 1 that the binomial tree of order 3k has a vertex-coloring with 2k+1 colors such that every path contains some color odd number of times. This disproves a conjecture from [1] asserting that for every tree T the minimal number of colors in a such coloring of T is at least the vertex ranking number of T minus one.

Let $\alpha \left(n\right)$ be the least number $k$ for which there exists a simple graph with $k$ vertices having precisely $n\ge 3$ spanning trees. Similarly, define $\beta \left(n\right)$ as the least number $k$ for which there exists a simple graph with $k$ edges having precisely $n\ge 3$ spanning trees. As an $n$-cycle has exactly $n$ spanning trees, it follows that $\alpha \left(n\right),\beta \left(n\right)\le n$. In this paper, we show that $\alpha \left(n\right)\le \frac{1}{3}(n+4)$ and $\beta \left(n\right)\le \frac{1}{3}(n+7)$ if and only if $n\notin \{3,4,5,6,7,9,10,13,18,22\}$, which is a subset of Euler’s idoneal numbers. Moreover, if $n\neg \equiv 2\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}3)$ and $n\ne 25$ we show that $\alpha \left(n\right)\le \frac{1}{4}(n+9)$ and $\beta \left(n\right)\le \frac{1}{4}(n+13).$ This improves some previously estabilished bounds.

A well-known theorem of Hajós claims that every graph with chromathic number greater than k can be constructed from disjoint copies of the complete graph ${K}_{k+1}$ by repeated application of three simple operations. This classical result has been extended in 1978 to colorings of hypergraphs by C. Benzaken and in 1996 to list-colorings of graphs by S. Gravier. In this note, we capture both variations to extend Hajós’ theorem to list-colorings of hypergraphs.

An acquaintance network is a social structure made up of a set of actors and the ties between them. These ties change dynamically as a consequence of incessant interactions between the actors. In this paper we introduce a social network model called the Interaction-Based (IB) model that involves well-known sociological principles. The connections between the actors and the strength of the connections are influenced by the continuous positive and negative interactions between the actors and, vice...

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