arXiv subjects

Carl Johan Casselgren

Publications and source records attributed to Carl Johan Casselgren.

25 records · Page 2Linked to original sources

arXiv2017

Some Results on Cyclic Interval Edge Colorings of Graphs

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 $Δ(G)\geq 4$ admits a cyclic interval $Δ(G)$-coloring if for every vertex $v$ the degree $d_G(v)$ satisfies either $d_G(v)\geq Δ(G)-2$ 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 ($r\geq 2$) 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.

math.CO↗

arXiv2017

Coloring graphs of various maximum degree from random lists

Let $G=G(n)$ be a graph on $n$ vertices with maximum degree $Δ=Δ(n)$. Assign to each vertex $v$ of $G$ a list $L(v)$ of colors by choosing each list independently and uniformly at random from all $k$-subsets of a color set $\mathcal{C}$ of size $σ= σ(n)$. Such a list assignment is called a \emph{random $(k,\mathcal{C})$-list assignment}. In this paper, we are interested in determining the asymptotic probability (as $n \to \infty$) of the existence of a proper coloring $φ$ of $G$, such that $φ(v) \in L(v)$ for every vertex $v$ of $G$, a so-called $L$-coloring. We give various lower bounds on $σ$, in terms of $n$, $k$ and $Δ$, which ensures that with probability tending to 1 as $n \to \infty$ there is an $L$-coloring of $G$. In particular, we show, for all fixed $k$ and growing $n$, that if $σ(n) = ω(n^{1/k^2} Δ^{1/k})$ and $Δ=O\left(n^{\frac{k-1}{k(k^3+ 2k^2 - k +1)}}\right)$, then the probability that $G$ has an $L$-coloring tends to 1 as $n \rightarrow \infty$. If $k\geq 2$ and $Δ= Ω(n^{1/2})$, then the same conclusion holds provided that $σ=ω(Δ)$. We also give related results for other bounds on $Δ$, when $k$ is constant or a strictly increasing function of $n$.

math.CO↗

arXiv2016

Some bounds on the number of colors in interval and cyclic interval edge colorings of graphs

An \emph{interval $t$-coloring} of a multigraph $G$ is a proper edge coloring with colors $1,\dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors. A \emph{cyclic interval $t$-coloring} of a multigraph $G$ is a proper edge coloring with colors $1,\dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors, under the condition that color $1$ is considered as consecutive to color $t$. Denote by $w(G)$ ($w_{c}(G)$) and $W(G)$ ($W_{c}(G)$) the minimum and maximum number of colors in a (cyclic) interval coloring of a multigraph $G$, respectively. We present some new sharp bounds on $w(G)$ and $W(G)$ for multigraphs $G$ satisfying various conditions. In particular, we show that if $G$ is a $2$-connected multigraph with an interval coloring, then $W(G)\leq 1+\left\lfloor \frac{|V(G)|}{2}\right\rfloor(Δ(G)-1)$. We also give several results towards the general conjecture that $W_{c}(G)\leq |V(G)|$ for any triangle-free graph $G$ with a cyclic interval coloring; we establish that approximate versions of this conjecture hold for several families of graphs, and we prove that the conjecture is true for graphs with maximum degree at most $4$.

math.CO↗

arXiv2016

Restricted completion of sparse partial Latin squares

An $n \times n$ partial Latin square $P$ is called $α$-dense if each row and column has at most $αn$ non-empty cells and each symbol occurs at most $αn$ times in $P$. An $n \times n$ array $A$ where each cell contains a subset of $\{1,\dots, n\}$ is a $(βn, βn, βn)$-array if each symbol occurs at most $βn$ times in each row and column and each cell contains a set of size at most $βn$. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants $α, β> 0$ such that, for every positive integer $n$, if $P$ is an $α$-dense $n \times n$ partial Latin square, $A$ is an $n \times n$ $(βn, βn, βn)$-array, and no cell of $P$ contains a symbol that appears in the corresponding cell of $A$, then there is a completion of $P$ that avoids $A$; that is, there is a Latin square $L$ that agrees with $P$ on every non-empty cell of $P$, and, for each $i,j$ satisfying $1 \leq i,j \leq n$, the symbol in position $(i,j)$ in $L$ does not appear in the corresponding cell of $A$.

math.CO↗

arXiv2014

Solution of Vizing's Problem on Interchanges for Graphs with Maximum Degree 4 and Related Results

Let $G$ be a Class 1 graph with maximum degree $4$ and let $t\geq 5$ be an integer. We show that any proper $t$-edge coloring of $G$ can be transformed to any proper $4$-edge coloring of $G$ using only transformations on $2$-colored subgraphs (so-called interchanges). This settles the smallest previously unsolved case of a well-known problem of Vizing on interchanges, posed in 1965. Using our result we give an affirmative answer to a question of Mohar for two classes of graphs: we show that all proper $5$-edge colorings of a Class 1 graph with maximum degree 4 are Kempe equivalent, that is, can be transformed to each other by interchanges, and that all proper 7-edge colorings of a Class 2 graph with maximum degree 5 are Kempe equivalent.

math.CO↗

arXiv2007

On path factors of (3,4)-biregular bigraphs

A (3,4)-biregular bigraph G is a bipartite graph where all vertices in one part have degree 3 and all vertices in the other part have degree 4. A path factor of G is a spanning subgraph whose components are nontrivial paths. We prove that a simple (3,4)-biregular bigraph always has a path factor such that the endpoints of each path have degree three. Moreover we suggest a polynomial algorithm for the construction of such a path factor.

math.CO↗

arXiv2007

Proper path-factors and interval edge-coloring of (3,4)-biregular bigraphs

An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3,4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2,4,6,8}. We provide sufficient conditions for the existence of such a subgraph.

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