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Combinatorics (math.CO)

Fri, 02 Jun 2023

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1.New bounds for odd colourings of graphs

Authors:Tianjiao Dai, Qiancheng Ouyang, François Pirot

Abstract: Given a graph $G$, a vertex-colouring $\sigma$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in \sigma(X)$ is said to be \emph{odd} for $X$ in $\sigma$ if it has an odd number of occurrences in $X$. We say that $\sigma$ is an \emph{odd colouring} of $G$ if it is proper and every (open) neighbourhood has an odd colour in $\sigma$. The odd chromatic number of a graph $G$, denoted by $\chi_o(G)$, is the minimum $k\in\mathbb{N}$ such that an odd colouring $\sigma \colon V(G)\to [k]$ exists. In a recent paper, Caro, Petru\v sevski and \v Skrekovski conjectured that every connected graph of maximum degree $\Delta\ge 3$ has odd-chromatic number at most $\Delta+1$. We prove that this conjecture holds asymptotically: for every connected graph $G$ with maximum degree $\Delta$, $\chi_o(G)\le\Delta+O(\ln\Delta)$ as $\Delta \to \infty$. We also prove that $\chi_o(G)\le\lfloor3\Delta/2\rfloor+2$ for every $\Delta$. If moreover the minimum degree $\delta$ of $G$ is sufficiently large, we have $\chi_o(G) \le \chi(G) + O(\Delta \ln \Delta/\delta)$ and $\chi_o(G) = O(\chi(G)\ln \Delta)$. Finally, given an integer $h\ge 1$, we study the generalisation of these results to $h$-odd colourings, where every vertex $v$ must have at least $\min \{\deg(v),h\}$ odd colours in its neighbourhood. Many of our results are tight up to some multiplicative constant.

2.Injective coloring of product graphs

Authors:Babak Samadi, Nasrin Soltankhah, Ismael G. Yero

Abstract: The problem of injective coloring in graphs can be revisited through two different approaches: coloring the two-step graphs and vertex partitioning of graphs into open packing sets, each of which is equivalent to the injective coloring problem itself. Taking these facts into account, we observe that the injective coloring lies between graph coloring and domination theory. We make use of these three points of view in this paper so as to investigate the injective coloring of some well-known graph products. We bound the injective chromatic number of direct and lexicographic product graphs from below and above. In particular, we completely determine this parameter for the direct product of two cycles. We also give a closed formula for the corona product of two graphs.

3.On Fan's conjecture about $4$-flow

Authors:Deping Song, Shuang Li, Xiao Wang

Abstract: Let $G$ be a bridgeless graph, $C$ is a circuit of $G$. Fan proposed a conjecture that if $G/C$ admits a nowhere-zero 4-flow, then $G$ admits a 4-flow $(D,f)$ such that $E(G)-E(C)\subseteq$ supp$(f)$ and $|\textrm{supp}(f)\cap E(C)|>\frac{3}{4}|E(C)|$. The purpose of this conjecture is to find shorter circuit cover in bridgeless graphs. Fan showed that the conjecture holds for $|E(C)|\le19.$ Wang, Lu and Zhang showed that the conjecture holds for $|E(C)|\le 27$. In this paper, we prove that the conjecture holds for $|E(C)|\le 35.$

4.Using alternating de Bruijn sequences to construct de Bruijn tori

Authors:Matthew Kreitzer, Mihai Nica, Rajesh Pereira

Abstract: A de Bruijn torus is the two dimensional generalization of a de Bruijn sequence. While some methods exist to generate these tori, only a few methods of construction are known. We present a novel method to generate de Bruijn tori with rectangular windows by combining two variants de Bruijn sequences called `Alternating de Bruijn sequences' and `De Bruijn families'.

5.Strong domination number of graphs from primary subgraphs

Authors:Saeid Alikhani, Nima Ghanbari, Michael A. Henning

Abstract: A set $D$ of vertices is a strong dominating set in a graph $G$, if for every vertex $x\in V(G) \setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x) \leq deg(y)$. The strong domination number $\gamma_{st}(G)$ of $G$ is the minimum cardinality of a strong dominating set in $G$. Let $G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,\ldots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, and identifying these two vertices, and thereafter continuing in this manner inductively. The graphs $G_1,\ldots ,G_k$ are the primary subgraphs of $G$. In this paper, we study the strong domination number of $K_r$-gluing of two graphs and investigate the strong domination number for some particular cases of graphs from their primary subgraphs.

6.Stability of Rose Window graphs

Authors:Milad Ahanjideh, István Kovács, Klavdija Kutnar

Abstract: A graph $\Gamma$ is said to be stable if for the direct product $\Gamma\times\mathbf{K}_2$, ${\rm Aut}(\Gamma \times \mathbf{K}_2)$ is isomorphic to ${\rm Aut}(\Gamma) \times \mathbb{Z}_2$; otherwise, it is called unstable. An unstable graph is called non-trivially unstable when it is not bipartite and no two vertices have the same neighborhood. Wilson described nine families of unstable Rose Window graphs and conjectured that these contain all non-trivially unstable Rose Window graphs (2008). In this paper we show that the conjecture is true.

7.Distinct eigenvalues of the Transposition graph

Authors:Elena V. Konstantinova, Artem Kravchuk

Abstract: Transposition graph $T_n$ is defined as a Cayley graph over the symmetric group generated by all transpositions. It is known that all eigenvalues of $T_n$ are integers. Moreover, zero is its eigenvalue for any $n\geqslant 4$. But the exact distribution of the spectrum of the graph $T_n$ is unknown. In this paper we prove that integers from the interval $[-\frac{n-4}{2}, \frac{n-4}{2}]$ lie in the spectrum of $T_n$ if $n \geqslant 19$.

8.Excluding Surfaces as Minors in Graphs

Authors:Dimitrios M. Thilikos, Sebastian Wiederrecht

Abstract: We introduce an annotated extension of treewidth that measures the contribution of a vertex set $X$ to the treewidth of a graph $G.$ This notion provides a graph distance measure to some graph property $\mathcal{P}$: A vertex set $X$ is a $k$-treewidth modulator of $G$ to $\mathcal{P}$ if the treewidth of $X$ in $G$ is at most $k$ and its removal gives a graph in $\mathcal{P}.$This notion allows for a version of the Graph Minors Structure Theorem (GMST) that has no need for apices and vortices: $K_k$-minor free graphs are those that admit tree-decompositions whose torsos have $c_{k}$-treewidth modulators to some surface of Euler-genus $c_{k}.$ This reveals that minor-exclusion is essentially tree-decomposability to a ``modulator-target scheme'' where the modulator is measured by its treewidth and the target is surface embeddability. We then fix the target condition by demanding that $\Sigma$ is some particular surface and define a ``surface extension'' of treewidth, where $\Sigma\mbox{-}\mathsf{tw}(G)$ is the minimum $k$ for which $G$ admits a tree-decomposition whose torsos have a $k$-treewidth modulator to being embeddable in $\Sigma.$We identify a finite collection $\mathfrak{D}_{\Sigma}$ of parametric graphs and prove that the minor-exclusion of the graphs in $\mathfrak{D}_{\Sigma}$ precisely determines the asymptotic behavior of ${\Sigma}\mbox{-}\mathsf{tw},$ for every surface $\Sigma.$ It follows that the collection $\mathfrak{D}_{\Sigma}$ bijectively corresponds to the ``surface obstructions'' for $\Sigma,$ i.e., surfaces that are minimally non-contained in $\Sigma.$

9.The diameter of randomly twisted hypercubes

Authors:Lucas Aragão, Maurício Collares, Gabriel Dahia, João Pedro Marciano

Abstract: The $n$-dimensional random twisted hypercube $\mathbf{G}_n$ is constructed recursively by taking two instances of $\mathbf{G}_{n-1}$, with any joint distribution, and adding a random perfect matching between their vertex sets. Benjamini, Dikstein, Gross, and Zhukovskii showed that its diameter is $O(n\log \log \log n/\log \log n)$ with high probability and at least ${(n - 1)/ \log_2 n}$. We improve their upper bound by showing that $$\operatorname{diam}(\mathbf{G}_n) = \big(1 + o(1)\big) \frac{n}{\log_2 n}$$ with high probability.