An improved upper bound on the adjacent vertex distinguishing total chromatic number of graphs

An adjacent vertex distinguishing total $k$-coloring of a graph $G$ is a proper total $k$-coloring of $G$ such that any pair of adjacent vertices have different sets of colors. The minimum number $k$ needed for such a total coloring of $G$ is denoted by ${\chi }_a^{\prime \prime }\left(G\right)$. In this paper we prove that ${\chi }_a^{\prime \prime }\left(G\right)\le 2\Delta \left(G\right)?1$ if $\Delta \left(G\right)\ge 4$, and ${\chi }_a^{\prime \prime }\left(G\right)\le ?\frac{5\Delta \left(G\right)+8}{3}?$ in general. This improves a result in Huang et al. (2012) which states that ${\chi }_a^{\prime \prime }\left(G\right)\le 2\Delta \left(G\right)$ for any graph with $\Delta \left(G\right)\ge 3$.     

A degree sum condition for graphs to be covered by two cycles

Let $G$ be a $k$-connected graph of order $n$. In [1], Bondy (1980) considered a degree sum condition for a graph to have a Hamiltonian cycle, say, to be covered by one cycle. He proved that if ${\sigma }_{k+1}\left(G\right)>(k+1)(n?1)/2$, then $G$ has a Hamiltonian cycle. On the other hand, concerning a degree sum condition for a graph to be covered by two cycles, Enomoto et al. (1995) [4] proved that if $k=1$ and ${\sigma }_3\left(G\right)\ge n$, then $G$ can be covered by two cycles. By these results, we conjecture that if ${\sigma }_{2k+1}\left(G\right)>(2k+1)(n?1)/3$, then $G$ can be covered by two cycles. In this paper, we prove the case $k=2$ of this conjecture. In fact, we prove a stronger result; if $G$ is 2-connected with ${\sigma }_5\left(G\right)\ge 5(n?1)/3$, then $G$ can be covered by two cycles, or $G$ belongs to an exceptional class.     

Sparse graphs of girth at least five are packable

A graph is packable if it is a subgraph of its complement. The following statement was conjectured by Faudree, Rousseau, Schelp and Schuster in 1981: every non-star graph $G$ with girth at least $5$ is packable.The conjecture was proved by Faudree et al. with the additional condition that $G$ has at most $\frac{6}{5}n?2$ edges. In this paper, for each integer $k\ge 3$, we prove that every non-star graph with girth at least $5$ and at most $\frac{2k?1}{k}n?{\alpha }_k\left(n\right)$ edges is packable, where ${\alpha }_k\left(n\right)$ is $o\left(n\right)$ for every $k$. This implies that the conjecture is true for sufficiently large planar graphs.     

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).     

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 $k\ge 2$.     

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 $2?{(k+1)}^{l+1}?2$.     

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.     

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 $G$.     

Monochromatic loose path partitions in k-uniform hypergraphs

A conjecture of Gyárfás and Sárközy says that in every $2$-coloring of the edges of the complete $k$-uniform hypergraph $K_n^k$, there are two disjoint monochromatic loose paths of distinct colors such that they cover all but at most $k?2$ vertices. A weaker form of this conjecture with $2k?5$ uncovered vertices instead of $k?2$ is proved. Thus the conjecture holds for $k=3$. The main result of this paper states that the conjecture is true for all $k\ge 3$.     

A characterization for the neighbor-distinguishing total chromatic number of planar graphs with Δ=13

The neighbor-distinguishing total chromatic number ${\chi }_a^{\prime \prime }\left(G\right)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be totally colored using $k$ colors with a condition that any two adjacent vertices have different sets of colors. In this paper, we give a sufficient and necessary condition for a planar graph $G$ with maximum degree 13 to have ${\chi }_a^{\prime \prime }\left(G\right)=14$ or ${\chi }_a^{\prime \prime }\left(G\right)=15$. Precisely, we show that if $G$ is a planar graph of maximum degree 13, then $14\le {\chi }_a^{\prime \prime }\left(G\right)\le 15$; and ${\chi }_a^{\prime \prime }\left(G\right)=15$ if and only if $G$ contains two adjacent 13-vertices.     

Extremal problems on the Hamiltonicity of claw-free graphs

In 1962, Erd?s proved that if a graph $G$ with $n$ vertices satisfies

$e\left(G\right)>max\left\{\left(\genfrac{}{}{0ex}{}{n?k}{2}\right)+k^2,\left(\genfrac{}{}{0ex}{}{?(n+1)∕2?}{2}\right)+{?\frac{n?1}{2}?}^2\right\},$

where the minimum degree $\delta \left(G\right)\ge k$ and $1\le k\le (n?1)∕2$, then it is Hamiltonian. For $n\ge 2k+1$, let $E_n^k=K_k\vee (k{K}_1+K_{n?2k})$, where “$\vee$” is the “join” operation. One can observe $e\left(E_n^k\right)=\left(\genfrac{}{}{0ex}{}{n?k}{2}\right)+k^2$ and $E_n^k$ is not Hamiltonian. As $E_n^k$ contains induced claws for $k\ge 2$, a natural question is to characterize all 2-connected claw-free non-Hamiltonian graphs with the largest possible number of edges. We answer this question completely by proving a claw-free analog of Erd?s’ theorem. Moreover, as byproducts, we establish several tight spectral conditions for a 2-connected claw-free graph to be Hamiltonian. Similar results for the traceability of connected claw-free graphs are also obtained. Our tools include Ryjá?ek’s claw-free closure theory and Brousek’s characterization of minimal 2-connected claw-free non-Hamiltonian graphs.     

Maximum matchings in regular graphs

It was conjectured by Mkrtchyan, Petrosyan and Vardanyan that every graph $G$ with $\Delta \left(G\right)?\delta \left(G\right)\le 1$ has a maximum matching $M$ such that any two $M$-unsaturated vertices do not share a neighbor. The results obtained in Mkrtchyan et al. (2010), Petrosyan (2014) and Picouleau (2010) leave the conjecture unknown only for $k$-regular graphs with $4\le k\le 6$. All counterexamples for $k$-regular graphs $(k\ge 7)$ given in Petrosyan (2014) have multiple edges. In this paper, we confirm the conjecture for all $k$-regular simple graphs and also $k$-regular multigraphs with $k\le 4$.     

List star chromatic index of sparse graphs

A star edge-coloring of a graph $G$ is a proper edge coloring such that every 2-colored connected subgraph of $G$ is a path of length at most 3. For a graph $G$, let the list star chromatic index of $G$, $c{h}_s^{\prime }\left(G\right)$, be the minimum $k$ such that for any $k$-uniform list assignment $L$ for the set of edges, $G$ has a star edge-coloring from $L$. Dvo?ák et al. (2013) asked whether the list star chromatic index of every subcubic graph is at most 7. In Kerdjoudj et al. (2017) we proved that it is at most 8. In this paper we consider graphs with any maximum degree, we proved that if the maximum average degree of a graph $G$ is less than $\frac{14}{5}$ (resp. 3), then $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+2$ (resp. $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+3$).     

Equation-regular sets and the Fox–Kleitman conjecture

Given $k\ge 1$, the Fox–Kleitman conjecture from 2006 states that there exists a nonzero integer $b$ such that the $2k$-variable linear Diophantine equation

$\sum _{i=1}^k(x_i?y_i)=b$

is $(2k?1)$-regular. This is best possible, since Fox and Kleitman showed that for all $b\ge 1$, this equation is not $2k$-regular. While the conjecture has recently been settled for all $k\ge 2$, here we focus on the case $k=3$ and determine the degree of regularity of the corresponding equation for all $b\ge 1$. In particular, this independently confirms the conjecture for $k=3$. We also briefly discuss the case $k=4$.     

Upper bounds for the sum of Laplacian eigenvalues of graphs

Let G be a graph with n vertices and $e(G)$ edges, and let ${\mu }_1(G)?{\mu }_2(G)???{\mu }_n(G)=0$ be the Laplacian eigenvalues of G. Let $S_k(G)={\sum }_{i=1}^k{\mu }_i(G)$, where $1?k?n$. Brouwer conjectured that $S_k(G)?e(G)+\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ for $1?k?n$. It has been shown in Haemers et al. [7] that the conjecture is true for trees. We give upper bounds for $S_k(G)$, and in particular, we show that the conjecture is true for unicyclic and bicyclic graphs.     

Strong list-chromatic index of subcubic graphs

A strong $k$-edge-coloring of a graph G is an edge-coloring with $k$ colors in which every color class is an induced matching. The strong chromatic index of $G$, denoted by ${\chi }_s^{\prime }\left(G\right)$, is the minimum $k$ for which $G$ has a strong $k$-edge-coloring. In 1985, Erd?s and Ne?et?il conjectured that ${\chi }_s^{\prime }\left(G\right)\le \frac{5}{4}\Delta {\left(G\right)}^2$, where $\Delta \left(G\right)$ is the maximum degree of $G$. When $G$ is a graph with maximum degree at most 3, the conjecture was verified independently by Andersen and Horák, Qing, and Trotter. In this paper, we consider the list version of strong edge-coloring. In particular, we show that every subcubic graph has strong list-chromatic index at most 11 and every planar subcubic graph has strong list-chromatic index at most 10.