Illegal dumping in Slovenia is still widespread, despite well-organized system of waste collection. Nelegalno odlaganje je v Sloveniji {e vedno precej raz{irjeno kljub urejenemu sistemu zbiranja odpadkov. BOGDAN BRICELJJanez Matos, Kri{tof O{tir, Jaka Kranjc, Attractiveness of roads for illegal dumping with regard to regional differences in SloveniaAttractiveness of roads for illegal dumping with regard to regional differences in Slovenia ABSTRACT: The first countrywide register of illegal dump sites in Slovenia was created in 2010 and 2011. Due to its extensiveness it allows in-depth analyses of factors that affect illegal waste disposal. Prior research has already proven the impact of roads on the incidence of illegal dumps, but in this paper we investigate if regional differences significantly influence its expression. We consider the existing landscape typing, which divides Slovenia into 14 landscape-ecological types. We find significant differences between landscape types in the attractiveness of roads for disposal. They are especially pronounced when comparing the attractiveness of individual road categories.

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A sequence is Thue or nonrepetitive if it does not contain a repetition of any length. We consider a generalization of this notion. A $j$-subsequence of a sequence $S$ is a subsequence in which two consecutive terms are at indices of difference $j$ in $S$. A $k$-Thue sequence is a sequence in which every $j$-subsequence, for $1\le j \le k$, is also Thue. It was conjectured that $k+2$ symbols are enough to construct an arbitrarily long $k$-Thue sequence and shown that the conjecture holds for $k \in \{2,3,5\}$. In this paper we present a construction of $k$-Thue sequences on $2k$ symbols, which improves the previous bound of $2k + 10\sqrt{k}$. Additionaly, we define cyclic $k$-Thue sequences and present a construction of such sequences of arbitrary lengths when $k=2$ using four symbols, with three exceptions. As a corollary, we obtain tight bounds for total Thue colorings of cycles. We conclude the paper with some open problems.

We consider extremal values of sum-Balaban index among graphs on n vertices. We determine that the upper bound for the minimum value of the sum-Balaban index is at most 4.47934 when n goes to infinity. For small values of n we determine the extremal graphs and we observe that they are similar to dumbbell graphs, in most cases having one extra edge added to the corresponding extreme for the usual Balaban index. We show that in the class of balanced dumbbell graphs, those with clique sizes 4 √ 2 log 1 + √ 2 √ n + o( √ n) have asymptotically the smallest value of sum-Balaban index. We pose several conjectures and problems regarding this topic.

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