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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.
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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 $chi(G)=3$.
A proper edge coloring of a graph $G$ with colors $1,2,dots,t$ is called a emph{cyclic interval $t$-coloring} if for each vertex $v$ of $G$ the edges incident to $v$ are colored by consecutive colors, under the condition that color $1$ is considered as consecutive to color $t$. We prove that a bipartite graph $G$ with even maximum degree $Delta(G)geq 4$ admits a cyclic interval $Delta(G)$-coloring if for every vertex $v$ the degree $d_G(v)$ satisfies either $d_G(v)geq Delta(G)-2$ or $d_G(v)leq 2$. We also prove that every Eulerian bipartite graph $G$ with maximum degree at most $8$ has a cyclic interval coloring. Some results are obtained for $(a,b)$-biregular graphs, that is, bipartite graphs with the vertices in one part all having degree $a$ and the vertices in the other part all having degree $b$; it has been conjectured that all these have cyclic interval colorings. We show that all $(4,7)$-biregular graphs as well as all $(2r-2,2r)$-biregular ($rgeq 2$) graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this settles in the affirmative, a conjecture of Petrosyan and Mkhitaryan.
A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is $(k-1)$-colorable. In this paper, we prove that for every fixed integer $kge 1$, there are only finitely many $k$-vertex-critical ($P_5$,gem)-free graphs and $(P_5,overline{P_3+P_2})$-free graphs. To prove the results we use a known structure theorem for ($P_5$,gem)-free graphs combined with properties of $k$-vertex-critical graphs. Moreover, we characterize all $k$-vertex-critical ($P_5$,gem)-free graphs and $(P_5,overline{P_3+P_2})$-free graphs for $k in {4,5}$ using a computer generation algorithm.
For a graph $H$ and a $k$-chromatic graph $F,$ if the Turan graph $T_{k-1}(n)$ has the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large enough), then $H$ is called $F$-Turan-good, or $k$-Turan-good for short if $F$ is $K_k.$ In this paper, we construct some new classes of $k$-Turan-good graphs and prove that $P_4$ and $P_5$ are $k$-Turan-good for $kge4.$
Building upon the notion of Gutman index $operatorname{SGut}(G)$, Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph $G$. The emph{Steiner Gutman $k$-index} $operatorname{SGut}_k(G)$ of $G$ is defined by $operatorname{SGut}_k(G)$ $=sum_{Ssubseteq V(G), |S|=k}left(prod_{vin S}deg_G(v)right) d_G(S)$, in which $d_G(S)$ is the Steiner distance of $S$ and $deg_G(v)$ is the degree of $v$ in $G$. In this paper, we derive new sharp upper and lower bounds on $operatorname{SGut}_k$, and then investigate the Nordhaus-Gaddum-type results for the parameter $operatorname{SGut}_k$. We obtain sharp upper and lower bounds of $operatorname{SGut}_k(G)+operatorname{SGut}_k(overline{G})$ and $operatorname{SGut}_k(G)cdot operatorname{SGut}_k(overline{G})$ for a connected graph $G$ of order $n$, $m$ edges and maximum degree $Delta$, minimum degree $delta$.
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