On Interval Non-Edge-Colorable Eulerian Multigraphs

An edge-coloring of a multigraph $G$ with colors $1,\ldots,t$ is called an interval $t$-coloring if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. In this note, we show that all Eulerian multigraphs with an odd number of edges have no interval coloring. We also give some methods for constructing of interval non-edge-colorable Eulerian multigraphs.Comment: 4 page

On interval edge-colorings of outerplanar graphs

An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an interval $t$-coloring if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if it has an interval $t$-coloring for some positive integer $t$. For an interval colorable graph $G$, the least value of $t$ for which $G$ has an interval $t$-coloring is denoted by $w(G)$. A graph $G$ is outerplanar if it can be embedded in the plane so that all its vertices lie on the same (unbounded) face. In this paper we show that if $G$ is a 2-connected outerplanar graph with $\Delta(G)=3$, then $G$ is interval colorable and \begin{center} w(G)=\left\{\begin{tabular}{ll} 3, & if | V(G)| is even, \ 4, & if | V(G)| is odd. \end{tabular}% \right. \end{center} We also give a negative answer to the question of Axenovich on the outerplanar triangulations.Comment: 9 pages, 3 figure

Interval edge colorings of some products of graphs

An edge coloring of a graph $G$ with colors $1,2,\ldots ,t$ is called an interval $t$-coloring if for each $i\in \{1,2,\ldots,t\}$ there is at least one edge of $G$ colored by $i$, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable, if there is an integer $t\geq 1$ for which $G$ has an interval $t$-coloring. Let $\mathfrak{N}$ be the set of all interval colorable graphs. In 2004 Kubale and Giaro showed that if $G,H\in \mathfrak{N}$, then the Cartesian product of these graphs belongs to $\mathfrak{N}$. Also, they formulated a similar problem for the lexicographic product as an open problem. In this paper we first show that if $G\in \mathfrak{N}$, then $G[nK_{1}]\in \mathfrak{N}$ for any $n\in \mathbf{N}$. Furthermore, we show that if $G,H\in \mathfrak{N}$ and $H$ is a regular graph, then strong and lexicographic products of graphs $G,H$ belong to $\mathfrak{N}$. We also prove that tensor and strong tensor products of graphs $G,H$ belong to $\mathfrak{N}$ if $G\in \mathfrak{N}$ and $H$ is a regular graph.Comment: 14 pages, 5 figures, minor change

Interval colorings of complete balanced multipartite graphs

A graph $G$ is called a complete $k$-partite ($k\geq 2$) graph if its vertices can be partitioned into $k$ independent sets $V_{1},...,V_{k}$ such that each vertex in $V_{i}$ is adjacent to all the other vertices in $V_{j}$ for $1\leq i<j\leq k$. A complete $k$-partite graph $G$ is a complete balanced $k$-partite graph if $|V_{1}| = |V_{2}| =... = |V_{k}|$. An edge-coloring of a graph $G$ with colors $1,...,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if $G$ has an interval $t$-coloring for some positive integer $t$. In this paper we show that a complete balanced $k$-partite graph $G$ with $n$ vertices in each part is interval colorable if and only if $nk$ is even. We also prove that if $nk$ is even and $(k-1)n\leq t\leq ((3/2)k-1)n-1$, then a complete balanced $k$-partite graph $G$ admits an interval $t$-coloring. Moreover, if $k=p2^{q}$, where $p$ is odd and $q\in \mathbb{N}$, then a complete balanced $k$-partite graph $G$ has an interval $t$-coloring for each positive integer $t$ satisfying $(k-1)n\leq t\leq (2k-p-q)n-1$.Comment: 10 page

Sequential edge-coloring on the subset of vertices of almost regular graphs

Let $G$ be a graph and $R\subseteq V(G)$. A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an $R$-sequential $t$-coloring if the edges incident to each vertex $v\in R$ are colored by the colors $1,\ldots,d_{G}(v)$, where $d_{G}(v)$ is the degree of the vertex $v$ in $G$. In this note, we show that if $G$ is a graph with $\Delta(G)-\delta(G)\leq 1$ and $\chi^{\prime}(G)=\Delta(G)=r$ ($r\geq 3$), then $G$ has an $R$-sequential $r$-coloring with $\vert R\vert \geq \left\lceil\frac{(r-1)n_{r}+n}{r}\right\rceil$, where $n=\vert V(G)\vert$ and $n_{r}=\vert\{v\in V(G):d_{G}(v)=r\}\vert$. As a corollary, we obtain the following result: if $G$ is a graph with $\Delta(G)-\delta(G)\leq 1$ and $\chi^{\prime}(G)=\Delta(G)=r$ ($r\geq 3$), then $\Sigma^{\prime}(G)\leq \left\lfloor\frac {2n_{r}(2r-1)+n(r-1)(r^{2}+2r-2)}{4r}\right\rfloor$, where $\Sigma^{\prime}(G)$ is the edge-chromatic sum of $G$.Comment: 4 page

On maximum matchings in almost regular graphs

In 2010, Mkrtchyan, Petrosyan and Vardanyan proved that every graph $G$ with $2\leq \delta(G)\leq \Delta(G)\leq 3$ contains a maximum matching whose unsaturated vertices do not have a common neighbor, where $\Delta(G)$ and $\delta(G)$ denote the maximum and minimum degrees of vertices in $G$, respectively. In the same paper they suggested the following conjecture: every graph $G$ with $\Delta(G)-\delta(G)\leq 1$ contains a maximum matching whose unsaturated vertices do not have a common neighbor. Recently, Picouleau disproved this conjecture by constructing a bipartite counterexample $G$ with $\Delta(G)=5$ and $\delta(G)=4$. In this note we show that the conjecture is false for graphs $G$ with $\Delta(G)-\delta(G)=1$ and $\Delta(G)\geq 4$, and for $r$-regular graphs when $r\geq 7$.Comment: 5 page

Interval Total Colorings of Complete Multipartite Graphs and Hypercubes

A total coloring of a graph $G$ is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. An interval total $t$-coloring of a graph $G$ is a total coloring of $G$ with colors $1,\ldots,t$ such that all colors are used, and the edges incident to each vertex $v$ together with $v$ are colored by $d_{G}(v)+1$ consecutive colors, where $d_{G}(v)$ is the degree of a vertex $v$ in $G$. In this paper we prove that all complete multipartite graphs with the same number of vertices in each part are interval total colorable. Moreover, we also give some bounds for the minimum and the maximum span in interval total colorings of these graphs. Next, we investigate interval total colorings of hypercubes $Q_{n}$. In particular, we prove that $Q_{n}$ ($n\geq 3$) has an interval total $t$-coloring if and only if $n+1\leq t\leq \frac{(n+1)(n+2)}{2}$.Comment: 17 page

Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift

We prove existence, uniqueness and optimal regularity of solutions to the stationary obstacle problem defined by the fractional Laplacian operator with drift, in the subcritical regime. We localize our problem by considering a suitable extension operator introduced by L. Caffarelli and L. Silvestre. The structure of the extension equation is different from the one considered by L. Caffarelli, S. Salsa and L. Silvestre in their study of the obstacle problem for the fractional Laplacian without drift, in that the obstacle function has less regularity, and exhibits some singularities. To take into account the new features of the problem, we prove a new Almgren-type monotonicity formula, which we then use to establish the optimal regularity of solutions

Interval edge-colorings of composition of graphs

An edge-coloring of a graph $G$ with consecutive integers $c_{1},\ldots,c_{t}$ is called an \emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if it has an interval $t$-coloring for some positive integer $t$. The set of all interval colorable graphs is denoted by $\mathfrak{N}$. In 2004, Giaro and Kubale showed that if $G,H\in \mathfrak{N}$, then the Cartesian product of these graphs belongs to $\mathfrak{N}$. In the same year they formulated a similar problem for the composition of graphs as an open problem. Later, in 2009, the first author showed that if $G,H\in \mathfrak{N}$ and $H$ is a regular graph, then $G[H]\in \mathfrak{N}$. In this paper, we prove that if $G\in \mathfrak{N}$ and $H$ has an interval coloring of a special type, then $G[H]\in \mathfrak{N}$. Moreover, we show that all regular graphs, complete bipartite graphs and trees have such a special interval coloring. In particular, this implies that if $G\in \mathfrak{N}$ and $T$ is a tree, then $G[T]\in \mathfrak{N}$.Comment: 12 pages, 3 figure

On sum edge-coloring of regular, bipartite and split graphs

An edge-coloring of a graph $G$ with natural numbers is called a sum edge-coloring if the colors of edges incident to any vertex of $G$ are distinct and the sum of the colors of the edges of $G$ is minimum. The edge-chromatic sum of a graph $G$ is the sum of the colors of edges in a sum edge-coloring of $G$. It is known that the problem of finding the edge-chromatic sum of an $r$-regular ($r\geq 3$) graph is $NP$-complete. In this paper we give a polynomial time $(1+\frac{2r}{(r+1)^{2}})$-approximation algorithm for the edge-chromatic sum problem on $r$-regular graphs for $r\geq 3$. Also, it is known that the problem of finding the edge-chromatic sum of bipartite graphs with maximum degree 3 is $NP$-complete. We show that the problem remains $NP$-complete even for some restricted class of bipartite graphs with maximum degree 3. Finally, we give upper bounds for the edge-chromatic sum of some split graphs.Comment: 11 page