The complete tripartite graph $K_{n,n,n}$ has $3n^2$ edges. For any collection of positive integers $x_1,x_2,\cdots ,x_m$ with ${\sum }_{i=1}^m{x}_i=3n^2$ and $x_i\ge 3$ for $1\le i\le m$, we exhibit an edge-disjoint decomposition of $K_{n,n,n}$ into closed trails (circuits) of lengths $x_1,x_2,\cdots ,x_m$.

Let $\mathrm{Lct}\left(G\right)$ denote the set of all lengths of closed trails that exist in an even graph $G$. A sequence $(t_1,\cdots ,t_p)$ of elements of $\mathrm{Lct}\left(G\right)$ adding up to $\left|E\right(G\left)\right|$ is $G$-realisable provided there is a sequence $(T_1,\cdots ,T_p)$ of pairwise edge-disjoint closed trails in $G$ such that $T_i$ is of length $t_i$ for $i=1,\cdots ,p$. The graph $G$ is arbitrarily decomposable into closed trails if all possible sequences are $G$-realisable. In the paper it is proved that if $a\ge 1$ is an odd integer and $M_{a,a}$ is a perfect matching in $K_{a,a}$, then the graph $K_{a,a}-M_{a,a}$ is arbitrarily decomposable into closed...

We give necessary and sufficient conditions for the decomposition of complete bipartite multigraph Km,n(λ) into paths and cycles having k edges. In particular, we show that such decomposition exists in Km,n(λ), when λ ≡ 0 (mod 2), [...] and k(p + q) = 2mn for k ≡ 0 (mod 2) and also when λ ≥ 3, λm ≡ λn ≡ 0(mod 2), k(p + q) =λ_mn, m, n ≥ k, (resp., m, n ≥ 3k/2) for k ≡ 0(mod 4) (respectively, for k ≡ 2(mod 4)). In fact, the necessary conditions given above are also sufficient when λ = 2.

In this note, we consider the problem of existence of an edge-decomposition of a multigraph into isomorphic copies of 2-edge paths $K_{1,2}$. We find necessary and sufficient conditions for such a decomposition of a multigraph H to exist when (i) either H does not have incident multiple edges or (ii) multiplicities of the edges in H are not greater than two. In particular, we answer a problem stated by Z. Skupień.

It is proved that a connected multigraph G which is the union of two edge-disjoint paths has another decomposition into two paths with the same set, U, of endvertices provided that the multigraph is neither a path nor cycle. Moreover, then the number of such decompositions is proved to be even unless the number is three, which occurs exactly if G is a tree homeomorphic with graph of either symbol + or ⊥. A multigraph on n vertices with exactly two traceable pairs is constructed for each n ≥ 3. The...

For n ≥ 4, the complete n-vertex multidigraph with arc multiplicity λ is proved to have a decomposition into directed paths of arbitrarily prescribed lengths ≤ n - 1 and different from n - 2, unless n = 5, λ = 1, and all lengths are to be n - 1 = 4. For λ = 1, a more general decomposition exists; namely, up to five paths of length n - 2 can also be prescribed.

In a given graph G = (V,E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a c ≥ χ(G) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number, denoted by d(G,c). The d(G = Cₘ × Kₙ, χ(G)) has been studied. In this note we show that the exact value of defining number d(G = Cₘ × Kₙ, c)...

Let $r\ge 3$, $n\ge r$ and $\pi =(d_1,d_2,...,d_n)$ be a non-increasing sequence of nonnegative integers. If $\pi$ has a realization $G$ with vertex set $V\left(G\right)=\{v_1,v_2,...,v_n\}$ such that $d_G\left(v_i\right)=d_i$ for $i=1,2,...,n$ and $v_1{v}_2\cdots v_r{v}_1$ is a cycle of length $r$ in $G$, then $\pi$ is said to be potentially $C_r^{\text{'}\text{'}}$-graphic. In this paper, we give a characterization for $\pi$ to be potentially $C_r^{\text{'}\text{'}}$-graphic.

The line graph of a graph $G$, denoted by $L\left(G\right)$, has $E\left(G\right)$ as its vertex set, where two vertices in $L\left(G\right)$ are adjacent if and only if the corresponding edges in $G$ have a vertex in common. For a graph $H$, define ${\overline{\sigma }}_2\left(H\right)=min\{d\left(u\right)+d\left(v\right):uv\in E\left(H\right)\}$. Let $H$ be a 2-connected claw-free simple graph of order $n$ with $\delta \left(H\right)\ge 3$. We show that, if ${\overline{\sigma }}_2\left(H\right)\ge \frac{1}{7}(2n-5)$ and $n$ is sufficiently large, then either $H$ is traceable or the Ryjáček’s closure $\mathrm{cl}\left(H\right)=L\left(G\right)$, where $G$ is an essentially $2$-edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10 which have...

A graph G of order n is called arbitrarily partitionable (AP for short) if, for every sequence (n1, . . . , nk) of positive integers with n1 + ⋯ + nk = n, there exists a partition (V1, . . . , Vk) of the vertex set V (G) such that Vi induces a connected subgraph of order ni for i = 1, . . . , k. In this paper we show that every connected graph G of order n ≥ 22 and with [...] ‖G‖ > (n−42)+12 $\left|\right|G\left|\right|>\left(n-42\right)+12$ edges is AP or belongs to few classes of exceptional graphs.

The nth detour chromatic number, χₙ(G) of a graph G is the minimum number of colours required to colour the vertices of G such that no path with more than n vertices is monocoloured. The number of vertices in a longest path of G is denoted by τ( G). We conjecture that χₙ(G) ≤ ⎡(τ(G))/n⎤ for every graph G and every n ≥ 1 and we prove results that support the conjecture. We also present some sufficient conditions for a graph to have nth chromatic number at most 2.