A degree condition for a graph to have (a,b)-parity factors

Let $a$ and $b$ be two positive integers such that $a\le b$ and $a\equiv b(mod2)$. A graph $F$ is an $(a,b)$-parity factor of a graph $G$ if $F$ is a spanning subgraph of $G$ and for all vertices $v\in V\left(F\right)$, $d_F\left(v\right)\equiv b(mod2)$ and $a\le d_F\left(v\right)\le b$. In this paper we prove that every connected graph $G$ with $n\ge b(a+b)(a+b+2)∕\left(2a\right)$ vertices has an $(a,b)$-parity factor if $na$ is even, $\delta \left(G\right)\ge (b?a)∕a+a$, and for any two nonadjacent vertices $u,v\in V\left(G\right)$, $max\{d_G\left(u\right),d_G\left(v\right)\}\ge \frac{an}{a+b}$. This extends an earlier result of Nishimura (1992) and strengthens a result of Cai and Li (1998).     

b-coloring of Kneser graphs

A $b$-coloring of a graph $G$ with $k$ colors is a proper coloring of $G$ using $k$ colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer $k$ for which $G$ has a $b$-coloring using $k$ colors is the $b$-chromatic number $b\left(G\right)$ of $G$. The $b$-spectrum $S_b\left(G\right)$ of a graph $G$ is the set of positive integers $k,\chi \left(G\right)\le k\le b\left(G\right)$, for which $G$ has a $b$-coloring using $k$ colors. A graph $G$ is $b$-continuous if $S_b\left(G\right)$ = the closed interval $[\chi \left(G\right),b\left(G\right)]$. In this paper, we obtain an upper bound for the $b$-chromatic number of some families of Kneser graphs. In addition we establish that $[\chi \left(G\right),n+k+1]?S_b\left(G\right)$ for the Kneser graph $G=K(2n+k,n)$ whenever $3\le n\le k+1$. We also establish the $b$-continuity of some families of regular graphs which include the family of odd graphs.     

Kirchhoff index in line,subdivision and total graphs of a regular graph

Let $G$ be a connected regular graph and $l\left(G\right)$, $s\left(G\right)$, $t\left(G\right)$ the line, subdivision, total graphs of $G$, respectively. In this paper, we derive formulae and lower bounds of the Kirchhoff index of $l\left(G\right)$, $s\left(G\right)$ and $t\left(G\right)$, respectively. In particular, we give special formulae for the Kirchhoff index of $l\left(G\right)$, $s\left(G\right)$ and $t\left(G\right)$, where $G$ is a complete graph $K_n$, a regular complete bipartite graph $K_{n,n}$ and a cycle $C_n$.     

Maximum average degree and relaxed coloring

We say a graph is $(d,d,\dots ,d,0,\dots ,0)$-colorable with $a$ of $d$’s and $b$ of $0$’s if $V\left(G\right)$ may be partitioned into $b$ independent sets $O_1,O_2,\dots ,O_b$ and $a$ sets $D_1,D_2,\dots ,D_a$ whose induced graphs have maximum degree at most $d$. The maximum average degree, $mad\left(G\right)$, of a graph $G$ is the maximum average degree over all subgraphs of $G$. In this note, for nonnegative integers $a,b$, we show that if $mad\left(G\right)<\frac{4}{3}a+b$, then $G$ is $(1_1,1_2,\dots ,1_a,0_1,\dots ,0_b)$-colorable.     

On maximum matchings in König–Egerváry graphs

For a graph $G$ let $\alpha \left(G\right),\mu \left(G\right)$, and $\tau \left(G\right)$ denote its independence number, matching number, and vertex cover number, respectively. If $\alpha \left(G\right)+\mu \left(G\right)=\left|V\left(G\right)\right|$ or, equivalently, $\mu \left(G\right)=\tau \left(G\right)$, then $G$ is a König–Egerváry graph.In this paper we give a new characterization of König–Egerváry graphs.     

Indicated coloring of graphs

We study a graph coloring game in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent realization of this project. The smallest number of colors necessary for Ann to win the game on a graph $G$ (regardless of Ben’s strategy) is called the indicated chromatic number of $G$, and denoted by ${\chi }_i\left(G\right)$. We approach the question how much ${\chi }_i\left(G\right)$ differs from the usual chromatic number $\chi \left(G\right)$. In particular, whether there is a function $f$ such that ${\chi }_i\left(G\right)?f\left(\chi \left(G\right)\right)$ for every graph $G$. We prove that $f$ cannot be linear with leading coefficient less than $4/3$. On the other hand, we show that the indicated chromatic number of random graphs is bounded roughly by $4\chi \left(G\right)$. We also exhibit several classes of graphs for which ${\chi }_i\left(G\right)=\chi \left(G\right)$ and show that this equality for any class of perfect graphs implies Clique-Pair Conjecture for this class of graphs.     

Restrained bondage in graphs

Let $G=(V,E)$ be a graph. A set $S?V$ is a restrained dominating set if every vertex not in $S$ is adjacent to a vertex in $S$ and to a vertex in $V?S$. The restrained domination number of $G$, denoted by ${\gamma }_r\left(G\right)$, is the smallest cardinality of a restrained dominating set of $G$. We define the restrained bondage number $b_r\left(G\right)$ of a nonempty graph $G$ to be the minimum cardinality among all sets of edges $E^{\prime }?E$ for which ${\gamma }_r(G?E^{\prime })>{\gamma }_r\left(G\right)$. Sharp bounds are obtained for $b_r\left(G\right)$, and exact values are determined for several classes of graphs. Also, we show that the decision problem for $b_r\left(G\right)$ is NP-complete even for bipartite graphs.     

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 $va\left(G\right)\le 2$.     

A note on the list vertex arboricity of toroidal graphs

The vertex arboricity $a\left(G\right)$ of a graph $G$ is the minimum number of colors required to color the vertices of $G$ such that no cycle is monochromatic. The list vertex arboricity $a_l\left(G\right)$ is the list-coloring version of this concept. In this note, we prove that if $G$ is a toroidal graph, then $a_l\left(G\right)\le 4$; and $a_l\left(G\right)=4$ if and only if $G$ contains $K_7$ as an induced subgraph.     

The edge spectrum of the saturation number for small paths

Let $H$ be a simple graph. A graph $G$ is called an $H$-saturated graph if $H$ is not a subgraph of $G$, but adding any missing edge to $G$ will produce a copy of $H$. Denote by $SAT(n,H)$ the set of all $H$-saturated graphs $G$ with order $n$. Then the saturation number $sat(n,H)$ is defined as ${min}_{G\in SAT(n,H)}\left|E\left(G\right)\right|$, and the extremal number $ex(n,H)$ is defined as ${max}_{G\in SAT(n,H)}\left|E\left(G\right)\right|$. A natural question is that of whether we can find an $H$-saturated graph with $m$ edges for any $sat(n,H)\le m\le ex(n,H)$. The set of all possible values $m$ is called the edge spectrum for $H$-saturated graphs. In this paper we investigate the edge spectrum for $P_i$-saturated graphs, where $2\le i\le 6$. It is trivial for the case of $P_2$ that the saturated graph must be an empty graph.     

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. $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+3$).     

Generalizing Tutte’s theorem and maximal non-matchable graphs

A graph $G$ has a perfect matching if and only if $0$ is not a root of its matching polynomial $\mu (G,x)$. Thus, Tutte’s famous theorem asserts that $0$ is not a root of $\mu (G,x)$ if and only if $c_{\mathrm{odd}}(G?S)\le \left|S\right|$ for all $S?V\left(G\right)$, where $c_{\mathrm{odd}}\left(G\right)$ denotes the number of odd components of $G$. Tutte’s theorem can be proved using a characterization of the structure of maximal non-matchable graphs, that is, the edge-maximal graphs among those having no perfect matching. In this paper, we prove a generalized version of Tutte’s theorem in terms of avoiding any given real number $\theta$ as a root of $\mu (G,x)$. We also extend maximal non-matchable graphs to maximal $\theta$-non-matchable graphs and determine the structure of such graphs.     

Essential edge connectivity of line graphs

Let $G$ be a graph and $L\left(G\right)$ be its line graph. In 1969, Chartrand and Stewart proved that ${\kappa }^{\prime }\left(L\left(G\right)\right)\ge 2{\kappa }^{\prime }\left(G\right)?2$, where ${\kappa }^{\prime }\left(G\right)$ and ${\kappa }^{\prime }\left(L\left(G\right)\right)$ denote the edge connectivity of $G$ and $L\left(G\right)$ respectively. We show a similar relationship holds for the essential edge connectivity of $G$ and $L\left(G\right)$, written ${\kappa }_e^{\prime }\left(G\right)$ and ${\kappa }_e^{\prime }\left(L\left(G\right)\right)$, respectively. In this note, it is proved that if $L\left(G\right)$ is not a complete graph and $G$ does not have a vertex of degree two, then ${\kappa }_e^{\prime }\left(L\left(G\right)\right)\ge 2{\kappa }_e^{\prime }\left(G\right)?2$. An immediate corollary is that $\kappa \left(L^2\left(G\right)\right)\ge 2\kappa \left(L\left(G\right)\right)?2$ for such graphs $G$, where the vertex connectivity of the line graph $L\left(G\right)$ and the second iterated line graph $L^2\left(G\right)$ are written as $\kappa \left(L\left(G\right)\right)$ and $\kappa \left(L^2\left(G\right)\right)$ respectively.     

Structure and algorithms for (cap,even hole)-free graphs

A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph $G$ has a vertex of degree at most $\frac{3}{2}\omega \left(G\right)?1$, and hence $\chi \left(G\right)\le \frac{3}{2}\omega \left(G\right)$, where $\omega \left(G\right)$ denotes the size of a largest clique in $G$ and $\chi \left(G\right)$ denotes the chromatic number of $G$. We give an $O\left(nm\right)$ algorithm for $q$-coloring these graphs for fixed $q$ and an $O\left(nm\right)$ algorithm for maximum weight stable set, where $n$ is the number of vertices and $m$ is the number of edges of the input graph. We also give a polynomial-time algorithm for minimum coloring.Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs $G$ without clique cutsets have treewidth at most $6\omega \left(G\right)?1$ and clique-width at most 48.