The NLC-width and clique-width for powers of graphs of bounded tree-width

The $k$-power graph of a graph $G$ is a graph with the same vertex set as $G$, in that two vertices are adjacent if and only if, there is a path between them in $G$ of length at most $k$. A $k$-tree-power graph is the $k$-power graph of a tree, a $k$-leaf-power graph is the subgraph of some $k$-tree-power graph induced by the leaves of the tree.We show that (1) every $k$-tree-power graph has NLC-width at most $k+2$ and clique-width at most $k+2+max\{?\frac{k}{2}??1,0\}$, (2) every $k$-leaf-power graph has NLC-width at most $k$ and clique-width at most $k+max\{?\frac{k}{2}??2,0\}$, and (3) every $k$-power graph of a graph of tree-width $l$ has NLC-width at most ${(k+1)}^{l+1}?1$, and clique-width at most $2?{(k+1)}^{l+1}?2$.     

A new approach to the chromatic number of the square of Kneser graph K(2k+1,k)

The vertices of Kneser graph $K(n,k)$ are the subsets of $\{1,2,\dots ,n\}$ of cardinality $k$, two vertices are adjacent if and only if they are disjoint. The square $G^2$ of a graph $G$ is defined on the vertex set of $G$ with two vertices adjacent if their distance in $G$ is at most 2. Z. Füredi, in 2002, proposed the problem of determining the chromatic number of the square of the Kneser graph. The first non-trivial problem arises when $n=2k+1$. It is believed that $\chi \left(K^2(2k+1,k)\right)=2k+c$ where $c$ is a constant, and yet the problem remains open. The best known upper bounds are by Kim and Park: $8k∕3+20∕3$ for 1$k\ge 3$ (Kim and Park, 2014) and $32k∕15+32$ for $k\ge 7$ (Kim and Park, 2016). In this paper, we develop a new approach to this coloring problem by employing graph homomorphisms, cartesian products of graphs, and linear congruences integrated with combinatorial arguments. These lead to $\chi \left(K^2(2k+1,k)\right)\le 5k∕2+c$, where $c$ is a constant in $\{5∕2,9∕2,5,6\}$, depending on $k\ge 2$.     

Stability in the Erdős–Gallai Theorem on cycles and paths,II

The Erd?s–Gallai Theorem states that for $k\ge 3$, any $n$-vertex graph with no cycle of length at least $k$ has at most $\frac{1}{2}(k?1)(n?1)$ edges. A stronger version of the Erd?s–Gallai Theorem was given by Kopylov: If $G$ is a 2-connected $n$-vertex graph with no cycle of length at least $k$, then $e\left(G\right)\le max\{h(n,k,2),h(n,k,?\frac{k?1}{2}?)\}$, where $h(n,k,a)?\left(\genfrac{}{}{0ex}{}{k?a}{2}\right)+a(n?k+a)$. Furthermore, Kopylov presented the two possible extremal graphs, one with $h(n,k,2)$ edges and one with $h(n,k,?\frac{k?1}{2}?)$ edges.In this paper, we complete a stability theorem which strengthens Kopylov’s result. In particular, we show that for $k\ge 3$ odd and all $n\ge k$, every $n$-vertex 2-connected graph $G$ with no cycle of length at least $k$ is a subgraph of one of the two extremal graphs or $e\left(G\right)\le max\{h(n,k,3),h(n,k,\frac{k?3}{2})\}$. The upper bound for $e\left(G\right)$ here is tight.     

On vertex-disjoint cycles and degree sum conditions

This paper considers a degree sum condition sufficient to imply the existence of $k$ vertex-disjoint cycles in a graph $G$. For an integer $t\ge 1$, let ${\sigma }_t\left(G\right)$ be the smallest sum of degrees of $t$ independent vertices of $G$. We prove that if $G$ has order at least $7k+1$ and ${\sigma }_4\left(G\right)\ge 8k?3$, with $k\ge 2$, then $G$ contains $k$ vertex-disjoint cycles. We also show that the degree sum condition on ${\sigma }_4\left(G\right)$ is sharp and conjecture a degree sum condition on ${\sigma }_t\left(G\right)$ sufficient to imply $G$ contains $k$ vertex-disjoint cycles for $k\ge 2$.     

On the vertex-arboricity of planar graphs without 7-cycles

The vertex arboricity $va\left(G\right)$ of a graph $G$ is the minimum number of colors the vertices can be labeled so that each color class induces a forest. It was well-known that $va\left(G\right)\le 3$ for every planar graph $G$. In this paper, we prove that $va\left(G\right)\le 2$ if $G$ is a planar graph without 7-cycles. This extends a result in [A. Raspaud, W. Wang, On the vertex-arboricity of planar graphs, European J. Combin. 29 (2008) 1064–1075] that for each $k\in \{3,4,5,6\}$, planar graphs $G$ without $k$-cycles have $va\left(G\right)\le 2$.     

Degree and neighborhood conditions for hamiltonicity of claw-free graphs

For a graph $H$, let ${\sigma }_t\left(H\right)=min\left\{{\Sigma }_{i=1}^t{d}_H\left(v_i\right)\right|\{v_1,v_2,\dots ,v_t\}\text{is an independent set in }H\}$ and let $U_t\left(H\right)=min\left\{\right|{?}_{i=1}^t{N}_H\left(v_i\right)\left|\right|\{v_1,v_2,?,v_t\}\text{is an independent set in }H\}$. We show that for a given number $?$ and given integers $p\ge t>0$, $k\in \{2,3\}$ and $N=N(p,?)$, if $H$ is a $k$-connected claw-free graph of order $n>N$ with $\delta \left(H\right)\ge 3$ and its Ryjác?ek’s closure $cl\left(H\right)=L\left(G\right)$, and if $d_t\left(H\right)\ge t(n+?)∕p$ where $d_t\left(H\right)\in \{{\sigma }_t\left(H\right),U_t\left(H\right)\}$, then either $H$ is Hamiltonian or $G$, the preimage of $L\left(G\right)$, can be contracted to a $k$-edge-connected $K_3$-free graph of order at most $max\{4p?5,2p+1\}$ and without spanning closed trails. As applications, we prove the following for such graphs $H$ of order $n$ with $n$ sufficiently large:(i) If $k=2$, $\delta \left(H\right)\ge 3$, and for a given $t$ ($1\le t\le 4$) $d_t\left(H\right)\ge \frac{tn}{4}$, then either $H$ is Hamiltonian or $cl\left(H\right)=L\left(G\right)$ where $G$ is a graph obtained from $K_{2,3}$ by replacing each of the degree 2 vertices by a $K_{1,s}$ ($s\ge 1$). When $t=4$ and $d_t\left(H\right)={\sigma }_4\left(H\right)$, this proves a conjecture in Frydrych (2001).(ii) If $k=3$, $\delta \left(H\right)\ge 24$, and for a given $t$ ($1\le t\le 10$) $d_t\left(H\right)>\frac{t(n+5)}{10}$, then $H$ is Hamiltonian. These bounds on $d_t\left(H\right)$ in (i) and (ii) are sharp. It unifies and improves several prior results on conditions involved ${\sigma }_t$ and $U_t$ for the hamiltonicity of claw-free graphs. Since the number of graphs of orders at most $max\{4p?5,2p+1\}$ are fixed for given $p$, improvements to (i) or (ii) by increasing the value of $p$ are possible with the help of a computer.     

Proper connection and size of graphs

An edge-coloured graph $G$ is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a connected graph $G,$ denoted by $\mathrm{pc}\left(G\right)$, is the smallest number of colours that are needed in order to make $G$ properly connected. Our main result is the following: Let $G$ be a connected graph of order $n$ and $k\ge 2$. If $\left|E\left(G\right)\right|\ge \left(\genfrac{}{}{0ex}{}{n?k?1}{2}\right)+k+2$, then $\mathrm{pc}\left(G\right)\le k$ except when $k=2$ and $G\in \{G_1,G_2\},$ where $G_1=K_1\vee (2K_1+K_2)$ and $G_2=K_1\vee (K_1+2K_2).$     

A sufficient condition for DP-4-colorability

DP-coloring of a simple graph is a generalization of list coloring, and also a generalization of signed coloring of signed graphs. It is known that for each $k\in \{3,4,5,6\}$, every planar graph without $C_k$ is 4-choosable. Furthermore, Jin et al. (2016) showed that for each $k\in \{3,4,5,6\}$, every signed planar graph without $C_k$ is signed 4-choosable. In this paper, we show that for each $k\in \{3,4,5,6\}$, every planar graph without $C_k$ is 4-DP-colorable, which is an extension of the above results.     

A note on degree sum conditions for 2-factors with a prescribed number of cycles in bipartite graphs

Let $G$ be a balanced bipartite graph of order $2n\ge 4$, and let ${\sigma }_{1,1}\left(G\right)$ be the minimum degree sum of two non-adjacent vertices in different partite sets of $G$. In 1963, Moon and Moser proved that if ${\sigma }_{1,1}\left(G\right)\ge n+1$, then $G$ is hamiltonian. In this note, we show that if $k$ is a positive integer, then the Moon–Moser condition also implies the existence of a 2-factor with exactly $k$ cycles for sufficiently large graphs. In order to prove this, we also give a ${\sigma }_{1,1}$ condition for the existence of $k$ vertex-disjoint alternating cycles with respect to a chosen perfect matching in $G$.     

Steiner tree packing number and tree connectivity

Let $S$ be a set of at least two vertices in a graph $G$. A subtree $T$ of $G$ is a $S$-Steiner tree if $S?V\left(T\right)$. Two $S$-Steiner trees $T_1$ and $T_2$ are edge-disjoint (resp. internally vertex-disjoint) if $E\left(T_1\right)\cap E\left(T_2\right)=?$ (resp. $E\left(T_1\right)\cap E\left(T_2\right)=?$ and $V\left(T_1\right)\cap V\left(T_2\right)=S$). Let ${\lambda }_G\left(S\right)$ (resp. ${\kappa }_G\left(S\right)$) be the maximum number of edge-disjoint (resp. internally vertex-disjoint) $S$-Steiner trees in $G$, and let ${\lambda }_k\left(G\right)$ (resp. ${\kappa }_k\left(G\right)$) be the minimum ${\lambda }_G\left(S\right)$ (resp. ${\kappa }_G\left(S\right)$) for $S$ ranges over all $k$-subset of $V\left(G\right)$. Kriesell conjectured that if ${\lambda }_G\left(\{x,y\}\right)\ge 2k$ for any $x,y\in S$, then ${\lambda }_G\left(S\right)\ge k$. He proved that the conjecture holds for $\left|S\right|=3,4$. In this paper, we give a short proof of Kriesell’s Conjecture for $\left|S\right|=3,4$, and also show that ${\lambda }_k\left(G\right)\ge ?\frac{1}{k?1}?\frac{k?}{2}??$ (resp. ${\kappa }_k\left(G\right)\ge ?\frac{1}{k?1}?\frac{k?}{2}??$ ) if $\lambda \left(G\right)\ge ?$ (resp. $\kappa \left(G\right)\ge ?$) in $G$, where $k=3,4$. Moreover, we also study the relation between ${\kappa }_k\left(L\left(G\right)\right)$ and ${\lambda }_k\left(G\right)$, where $L\left(G\right)$ is the line graph of $G$.     

On 3-stable number conditions in n-connected claw-free graphs

For a subgraph $X$ of $G$, let ${\alpha }_G^3\left(X\right)$ be the maximum number of vertices of $X$ that are pairwise distance at least three in $G$. In this paper, we prove three theorems. Let $n$ be a positive integer, and let $H$ be a subgraph of an $n$-connected claw-free graph $G$. We prove that if $n\ge 2$, then either $H$ can be covered by a cycle in $G$, or there exists a cycle $C$ in $G$ such that ${\alpha }_G^3(H?V\left(C\right))\le {\alpha }_G^3\left(H\right)?n$. This result generalizes the result of Broersma and Lu that $G$ has a cycle covering all the vertices of $H$ if ${\alpha }_G^3\left(H\right)\le n$. We also prove that if $n\ge 1$, then either $H$ can be covered by a path in $G$, or there exists a path $P$ in $G$ such that ${\alpha }_G^3(H?V\left(P\right))\le {\alpha }_G^3\left(H\right)?n?1$. By using the second result, we prove the third result. For a tree $T$, a vertex of $T$ with degree one is called a leaf of $T$. For an integer $k\ge 2$, a tree which has at most $k$ leaves is called a $k$-ended tree. We prove that if ${\alpha }_G^3\left(H\right)\le n+k?1$, then $G$ has a $k$-ended tree covering all the vertices of $H$. This result gives a positive answer to the conjecture proposed by Kano et al. (2012).