# Transactions on Combinatorics Volume:13 Issue: 4, Dec 2024

• تاریخ انتشار: 1403/09/11
• تعداد عناوین: 7
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• John Burke *, Maitland Burke, Leonardo Pinheiro, Cameron Richer Pages 305-317
We define slopes in the flat torus as the set of equivalence classes of the solutions of linear equations in $\mathbb{R}^2$. The definition is equivalent to that of closed geodesics in the flat torus passing through the equivalence class of the point $(0,0)$. In this paper we derive formulas for counting the number of points in the intersection of multiple slopes in the flat torus.
Keywords: Torus, intersection of curves, counting methods
• Shahram Mohsenipour * Pages 319-325
We give a purely combinatorial proof for the infinitary van der Waerden's theorem.
Keywords: arithmetic progressions, Colorings, Combinatorial proofs, Infinite version
• Bouroubi Sadek * Pages 327-334
Given an integer $n\geq4$, how many inequivalent quadrilaterals with ordered integer sides and perimeter $n$ are there? Denoting such number by $Q(n)$, the answer is given by the following closed formula:$Q(n)=\left\{ \dfrac{1}{576}n\left( n+3\right) \left( 2n+3\right) -\dfrac{\left( -1\right) ^{n}}{192}n\left( n-5\right) \right\} \cdot$
A $b$-coloring of a graph\ $G$ is a proper coloring of its vertices such that each color class contains a vertex that has a neighbor in every other color classes. The $b$-chromatic number of a graph $G$, denoted by $b(G)$, is the largest integer $k$ such that $G$ admits a $b$-coloring with $k$ colors. Let $G_{e}$ be the graph obtained from $G$ by subdividing the edge $e$. A graph $G$ is $sd_{b}$-critical if $b(G_{e})<b(G)$ holds for any edge $e$ of $G$. In this paper, we first present \ several basic properties of $sd_{b}$-critical graphs and then we give a characterization of $sd_{b}$-critical $P_{4}$-sparse graphs and $sd_{b}$-critical quasi-line graphs.
Keywords: $b$-coloring, $b$-chromatic number, $sd, {b}$-critical graphs
Let $n_1, n_2,\ldots,n_k$ be integers and $V_1, V_2,\ldots,V_k$ be disjoint vertex sets with $|V_i|=n_i$ for each $i= 1, 2,\ldots,k$. A $k$-partite $k$-uniform hypergraph on vertex classes $V_1, V_2,\ldots,V_k$ is defined to be the $k$-uniform hypergraph whose edge set consists of the $k$-element subsets $S$ of $V_1 \cup V_2 \cup \cdots \cup V_k$ such that $|S\cap V_i|=1$ for all $i= 1, 2,\ldots,k$. We say that it is balanced if $n_1=n_2=\cdots=n_k$. In this paper, we give a distance spectral radius condition to guarantee the existence of perfect matching in $k$-partite $k$-uniform hypergraphs, this result generalize the result of Zhang and Lin  [Perfect matching and distance spectral radius in graphs and bipartite graphs, Discrete Appl. Math., 304 (2021) 315-322].
Keywords: $k$-uniform $k$-partite hypergraphs, Distance spectral radius, perfect matching