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Register a new userSome extremal results on the chromatic-stability index
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The $chi$-stability index ${rm es}_{chi}(G)$ of a graph $G$ is the minimum number of its edges whose removal results in a graph with the chromatic number smaller than that of $G$. In this paper three open problems from [European J. Combin. 84 (2020) 103042] are considered. Examples are constructed which demonstrate that a known characterization of $k$-regular ($kle 5$) graphs $G$ with ${rm es}_{chi}(G) = 1$ does not extend to $kge 6$. Graphs $G$ with $chi(G)=3$ for which ${rm es}_{chi}(G)+{rm es}_{chi}(overline{G}) = 2$ holds are characterized. Necessary conditions on graphs $G$ which attain a known upper bound on ${rm es}_{chi}(G)$ in terms of the order and the chromatic number of $G$ are derived. The conditions are proved to be sufficient when $nequiv 2 pmod 3$ and $chi(G)=3$.
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Given a proper edge coloring $varphi$ of a graph $G$, we define the palette $S_{G}(v,varphi)$ of a vertex $v in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $check s(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. In this paper we give various upper and lower bounds on the palette index of $G$ in terms of the vertex degrees of $G$, particularly for the case when $G$ is a bipartite graph with small vertex degrees. Some of our results concern $(a,b)$-biregular graphs; that is, bipartite graphs where all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. We conjecture that if $G$ is $(a,b)$-biregular, then $check{s}(G)leq 1+max{a,b}$, and we prove that this conjecture holds for several families of $(a,b)$-biregular graphs. Additionally, we characterize the graphs whose palette index equals the number of vertices.
In this paper, we characterize the extremal digraphs with the maximal or minimal $alpha$-spectral radius among some digraph classes such as rose digraphs, generalized theta digraphs and tri-ring digraphs with given size $m$. These digraph classes are denoted by $mathcal{R}_{m}^k$, $widetilde{boldsymbol{Theta}}_k(m)$ and $INF(m)$ respectively. The main results about spectral extremal digraph by Guo and Liu in cite{MR2954483} and Li and Wang in cite{MR3777498} are generalized to $alpha$-spectral graph theory. As a by-product of our main results, an open problem in cite{MR3777498} is answered. Furthermore, we determine the digraphs with the first three minimal $alpha$-spectral radius among all strongly connected digraphs. Meanwhile, we determine the unique digraph with the fourth minimal $alpha$-spectral radius among all strongly connected digraphs for $0le alpha le frac{1}{2}$.
The strong chromatic index of a graph $G$, denoted $chi_s(G)$, is the least number of colors needed to edge-color $G$ so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted $chi_{s,ell}(G)$, is the least integer $k$ such that if arbitrary lists of size $k$ are assigned to each edge then $G$ can be edge-colored from those lists where edges at distance at most two receive distinct colors. We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if $G$ is a subcubic planar graph with $operatorname{girth}(G) geq 41$ then $chi_{s,ell}(G) leq 5$, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759--770]. We further show that if $G$ is a subcubic planar graph and $operatorname{girth}(G) geq 30$, then $chi_s(G) leq 5$, improving a bound from the same paper. Finally, if $G$ is a planar graph with maximum degree at most four and $operatorname{girth}(G) geq 28$, then $chi_s(G) leq 7$, improving a more general bound of Wang and Zhao from [Odd graphs and its application on the strong edge coloring, arXiv:1412.8358] in this case.
In this note we obtain a new bound for the acyclic edge chromatic number $a(G)$ of a graph $G$ with maximum degree $D$ proving that $a(G)leq 3.569(D-1)$. To get this result we revisit and slightly modify the method described in [Giotis, Kirousis, Psaromiligkos and Thilikos, Theoretical Computer Science, 66: 40-50, 2017].
An extension of the well-known Szeged index was introduced recently, named as weighted Szeged index ($textrm{sz}(G)$). This paper is devoted to characterizing the extremal trees and graphs of this new topological invariant. In particular, we proved that the star is a tree having the maximal $textrm{sz}(G)$. Finding a tree with the minimal $textrm{sz}(G)$ is not an easy task to be done. Here, we present the minimal trees up to 25 vertices obtained by computer and describe the regularities which retain in them. Our preliminary computer tests suggest that a tree with the minimal $textrm{sz}(G)$ is also the connected graph of the given order that attains the minimal weighted Szeged index. Additionally, it is proven that among the bipartite connected graphs the complete balanced bipartite graph $K_{leftlfloor n/2rightrfloorleftlceil n/2 rightrceil}$ attains the maximal $textrm{sz}(G)$,. We believe that the $K_{leftlfloor n/2rightrfloorleftlceil n/2 rightrceil}$ is a connected graph of given order that attains the maximum $textrm{sz}(G)$.
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