critical graphs
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2022 ◽  
Vol 345 (3) ◽  
pp. 112717
Author(s):  
Muhuo Liu ◽  
Xiaofeng Gu
Keyword(s):  

2022 ◽  
Vol 100 ◽  
pp. 103451
Author(s):  
Jiaao Li ◽  
Yulai Ma ◽  
Yongtang Shi ◽  
Weifan Wang ◽  
Yezhou Wu
Keyword(s):  

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Benjamin R. Moore ◽  
Douglas B. West
Keyword(s):  

Tuza [1992] proved that a graph with no cycles of length congruent to $1$ modulo $k$ is $k$-colorable.  We prove that if a graph $G$ has an edge $e$ such that $G-e$ is $k$-colorable and $G$ is not, then for $2\le r\le k$, the edge $e$ lies in at least $\prod_{i=1}^{r-1} (k-i)$ cycles of length $1\mod r$ in $G$, and $G-e$ contains at least $\frac12{\prod_{i=1}^{r-1} (k-i)}$ cycles of length $0 \mod r$. A $(k,d)$-coloring of $G$ is a homomorphism from $G$ to the graph $K_{k:d}$ with vertex set ${\mathbb Z}_{k}$ defined by making $i$ and $j$ adjacent if $d\le j-i \le k-d$.  When $k$ and $d$ are relatively prime, define $s$ by $sd\equiv 1\mod k$.  A result of Zhu [2002] implies that $G$ is $(k,d)$-colorable when $G$ has no cycle $C$ with length congruent to $is$ modulo $k$ for any $i\in \{1,\ldots,2d-1\}$.  In fact, only $d$ classes need be excluded: we prove that if $G-e$ is $(k,d)$-colorable and $G$ is not, then $e$ lies in at least one cycle with length congruent to $is\mod k$ for some $i$ in $\{1,\ldots,d\}$.  Furthermore, if this does not occur with $i\in\{1,\ldots,d-1\}$, then $e$ lies in at least two cycles with length $1\mod k$ and $G-e$ contains a cycle of length $0 \mod k$.


2021 ◽  
Vol 344 (12) ◽  
pp. 112604
Author(s):  
Gang Chen ◽  
Zhengke Miao ◽  
Zi-Xia Song ◽  
Jingmei Zhang

Author(s):  
Rihab Hamid ◽  
Nour El Houda Bendahib ◽  
Mustapha Chellali ◽  
Nacéra Meddah

Let [Formula: see text] be a function on a graph [Formula: see text]. A vertex [Formula: see text] with [Formula: see text] is said to be undefended with respect to [Formula: see text] if it is not adjacent to a vertex [Formula: see text] with [Formula: see text]. A function [Formula: see text] is called a weak Roman dominating function (WRDF) if each vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text], such that the function [Formula: see text] defined by [Formula: see text], [Formula: see text] and [Formula: see text] for all [Formula: see text], has no undefended vertex. The weight of a WRDF is the sum of its function values over all vertices, and the weak Roman domination number [Formula: see text] is the minimum weight of a WRDF in [Formula: see text]. In this paper, we consider the effects of edge deletion on the weak Roman domination number of a graph. We show that the deletion of an edge of [Formula: see text] can increase the weak Roman domination number by at most 1. Then we give a necessary condition for [Formula: see text]-ER-critical graphs, that is, graphs [Formula: see text] whose weak Roman domination number increases by the deletion of any edge. Restricted to the class of trees, we provide a constructive characterization of all [Formula: see text]-ER-critical trees.


Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2568
Author(s):  
Norah Almalki ◽  
Pawaton Kaemawichanurat

A graph G is said to be k-γc-critical if the connected domination number γc(G) is equal to k and γc(G+uv)<k for any pair of non-adjacent vertices u and v of G. Let ζ be the number of cut vertices of G and let ζ0 be the maximum number of cut vertices that can be contained in one block. For an integer ℓ≥0, a graph G is ℓ-factor critical if G−S has a perfect matching for any subset S of vertices of size ℓ. It was proved by Ananchuen in 2007 for k=3, Kaemawichanurat and Ananchuen in 2010 for k=4 and by Kaemawichanurat and Ananchuen in 2020 for k≥5 that every k-γc-critical graph has at most k−2 cut vertices and the graphs with maximum number of cut vertices were characterized. In 2020, Kaemawichanurat and Ananchuen proved further that, for k≥4, every k-γc-critical graphs satisfies the inequality ζ0(G)≤mink+23,ζ. In this paper, we characterize all k-γc-critical graphs having k−3 cut vertices. Further, we establish realizability that, for given k≥4, 2≤ζ≤k−2 and 2≤ζ0≤mink+23,ζ, there exists a k-γc-critical graph with ζ cut vertices having a block which contains ζ0 cut vertices. Finally, we proved that every k-γc-critical graph of odd order with minimum degree two is 1-factor critical if and only if 1≤k≤2. Further, we proved that every k-γc-critical K1,3-free graph of even order with minimum degree three is 2-factor critical if and only if 1≤k≤2.


2021 ◽  
Vol 406 ◽  
pp. 126248
Author(s):  
Michael A. Henning ◽  
Pawaton Kaemawichanurat
Keyword(s):  

Author(s):  
Laurent Beaudou ◽  
Florent Foucaud ◽  
Reza Naserasr
Keyword(s):  

Author(s):  
Zhen-Kun Zhang ◽  
Zhong Zhao ◽  
Liu-Yong Pang
Keyword(s):  

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