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A proper edge coloring of a graph $G$ with colors $1,2,dots,t$ is called a 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 $t$. In this paper we introduce and investigate a new notion, the cyclic deficiency of a graph $G$, defined as the minimum number of pendant edges whose attachment to $G$ yields a graph admitting a cyclic interval coloring; this number can be considered as a measure of closeness of $G$ of being cyclically interval colorable. We determine or bound the cyclic deficiency of several families of graphs. In particular, we present examples of graphs of bounded maximum degree with arbitrarily large cyclic deficiency, and graphs whose cyclic deficiency approaches the number of vertices. Finally, we conjecture that the cyclic deficiency of any graph does not exceed the number of vertices, and we present several results supporting this conjecture.
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Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property $mathcal P$ and a graph $G$, the deficiency $text{def}(G)$ of the graph $G$ with respect to the property $mathcal P$ is the smallest non-negative integer $t$ such that the join $G*K_t$ has property $mathcal P$. In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an $n$-vertex graph $G$ needs to ensure $G*K_t$ contains a $K_r$-factor (for any fixed $rgeq 3$). In this paper we resolve their problem fully. We also give an analogous result which forces $G*K_t$ to contain any fixed bipartite $(n+t)$-vertex graph of bounded degree and small bandwidth.
An edge-coloring of a graph $G$ with colors $1,ldots,t$ is an emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an integer interval. It is well-known that there are graphs that do not have interval colorings. The emph{deficiency} of a graph $G$, denoted by $mathrm{def}(G)$, is the minimum number of pendant edges whose attachment to $G$ leads to a graph admitting an interval coloring. In this paper we investigate the problem of determining or bounding of the deficiency of complete multipartite graphs. In particular, we obtain a tight upper bound for the deficiency of complete multipartite graphs. We also determine or bound the deficiency for some classes of complete multipartite graphs.
A emph{proper $t$-edge-coloring} of a graph $G$ is a mapping $alpha: E(G)rightarrow {1,ldots,t}$ such that all colors are used, and $alpha(e) eq alpha(e^{prime})$ for every pair of adjacent edges $e,e^{prime}in E(G)$. If $alpha $ is a proper edge-coloring of a graph $G$ and $vin V(G)$, then emph{the spectrum of a vertex $v$}, denoted by $Sleft(v,alpha right)$, is the set of all colors appearing on edges incident to $v$. emph{The deficiency of $alpha$ at vertex $vin V(G)$}, denoted by $def(v,alpha)$, is the minimum number of integers which must be added to $Sleft(v,alpha right)$ to form an interval, and emph{the deficiency $defleft(G,alpharight)$ of a proper edge-coloring $alpha$ of $G$} is defined as the sum $sum_{vin V(G)}def(v,alpha)$. emph{The deficiency of a graph $G$}, denoted by $def(G)$, is defined as follows: $def(G)=min_{alpha}defleft(G,alpharight)$, where minimum is taken over all possible proper edge-colorings of $G$. For a graph $G$, the smallest and the largest values of $t$ for which it has a proper $t$-edge-coloring $alpha$ with deficiency $def(G,alpha)=def(G)$ are denoted by $w_{def}(G)$ and $W_{def}(G)$, respectively. In this paper, we obtain some bounds on $w_{def}(G)$ and $W_{def}(G)$. In particular, we show that for any $lin mathbb{N}$, there exists a graph $G$ such that $def(G)>0$ and $W_{def}(G)-w_{def}(G)geq l$. It is known that for the complete graph $K_{2n+1}$, $def(K_{2n+1})=n$ ($nin mathbb{N}$). Recently, Borowiecka-Olszewska, Drgas-Burchardt and Ha{l}uszczak posed the following conjecture on the deficiency of near-complete graphs: if $nin mathbb{N}$, then $def(K_{2n+1}-e)=n-1$. In this paper, we confirm this conjecture.
A proper edge-coloring of a graph $G$ with colors $1,ldots,t$ is called an emph{interval cyclic $t$-coloring} if all colors are used, and the edges incident to each vertex $vin V(G)$ are colored by $d_{G}(v)$ consecutive colors modulo $t$, where $d_{G}(v)$ is the degree of a vertex $v$ in $G$. A graph $G$ is emph{interval cyclically colorable} if it has an interval cyclic $t$-coloring for some positive integer $t$. The set of all interval cyclically colorable graphs is denoted by $mathfrak{N}_{c}$. For a graph $Gin mathfrak{N}_{c}$, the least and the greatest values of $t$ for which it has an interval cyclic $t$-coloring are denoted by $w_{c}(G)$ and $W_{c}(G)$, respectively. In this paper we investigate some properties of interval cyclic colorings. In particular, we prove that if $G$ is a triangle-free graph with at least two vertices and $Gin mathfrak{N}_{c}$, then $W_{c}(G)leq vert V(G)vert +Delta(G)-2$. We also obtain bounds on $w_{c}(G)$ and $W_{c}(G)$ for various classes of graphs. Finally, we give some methods for constructing of interval cyclically non-colorable graphs.
By complexity of a finite graph we mean the number of spanning trees in the graph. The aim of the present paper is to give a new approach for counting complexity $tau(n)$ of cyclic $n$-fold coverings of a graph. We give an explicit analytic formula for $tau(n)$ in terms of Chebyshev polynomials and find its asymptotic behavior as $ntoinfty$ through the Mahler measure of the associated voltage polynomial. We also prove that $F(x)=sumlimits_{n=1}^inftytau(n)x^n$ is a rational function with integer coefficients.
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